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85-469: APV may refer to: Actuarial present value , a probability weighted present value often used in insurance Adjusted present value , a variation of the net present value (NPV) Advanced Power Virtualization (renamed PowerVM ), a software virtualization technique used by IBM Alavuden Peli-Veikot , a multi-sport club in Alavus, Finland Allen Parkway Village ,

170-569: A n ( x ) {\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)} is a discrete distribution function. Here δ t ( x ) = 0 {\displaystyle \delta _{t}(x)=0} for x < t {\displaystyle x<t} , δ t ( x ) = 1 {\displaystyle \delta _{t}(x)=1} for x ≥ t {\displaystyle x\geq t} . Taking for instance an enumeration of all rational numbers as {

255-478: A n } {\displaystyle \{a_{n}\}} , one gets a discrete function that is not necessarily a step function (piecewise constant). The possible outcomes for one coin toss can be described by the sample space Ω = { heads , tails } {\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}} . We can introduce a real-valued random variable Y {\displaystyle Y} that models

340-394: A , b ] = { x ∈ R : a ≤ x ≤ b } {\textstyle I=[a,b]=\{x\in \mathbb {R} :a\leq x\leq b\}} , a random variable X I ∼ U ⁡ ( I ) = U ⁡ [ a , b ] {\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} [a,b]}

425-433: A mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die ; it may also represent uncertainty, such as measurement error . However,

510-417: A joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space Ω {\displaystyle \Omega } altogether and just puts a measure on R {\displaystyle \mathbb {R} } that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See

595-430: A probability density function , which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous. Any random variable can be described by its cumulative distribution function , which describes the probability that the random variable will be less than or equal to

680-457: A probability measure space (called the sample space ) to a measurable space . This allows consideration of the pushforward measure , which is called the distribution of the random variable; the distribution is thus a probability measure on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be independent . It

765-537: A probability space and ( E , E ) {\displaystyle (E,{\mathcal {E}})} a measurable space . Then an ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} , which means that, for every subset B ∈ E {\displaystyle B\in {\mathcal {E}}} , its preimage

850-493: A random variable is taken to be automatically valued in the real numbers, with more general random quantities instead being called random elements . According to George Mackey , Pafnuty Chebyshev was the first person "to think systematically in terms of random variables". A random variable X {\displaystyle X} is a measurable function X : Ω → E {\displaystyle X\colon \Omega \to E} from

935-404: A random variable . In this case the observation space is the set of real numbers. Recall, ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is the probability space. For a real observation space, the function X : Ω → R {\displaystyle X\colon \Omega \rightarrow \mathbb {R} }

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1020-426: A random variable of type E {\displaystyle E} , or an E {\displaystyle E} -valued random variable . This more general concept of a random element is particularly useful in disciplines such as graph theory , machine learning , natural language processing , and other fields in discrete mathematics and computer science , where one is often interested in modeling

1105-427: A $ 1 payoff for a successful bet on heads as follows: Y ( ω ) = { 1 , if  ω = heads , 0 , if  ω = tails . {\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\[6pt]0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}} If

1190-615: A CURV X ∼ U ⁡ [ a , b ] {\displaystyle X\sim \operatorname {U} [a,b]} is given by the indicator function of its interval of support normalized by the interval's length: f X ( x ) = { 1 b − a , a ≤ x ≤ b 0 , otherwise . {\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}} Of particular interest

1275-429: A certain value. The term "random variable" in statistics is traditionally limited to the real-valued case ( E = R {\displaystyle E=\mathbb {R} } ). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function , and the moments of its distribution. However,

1360-497: A continuous random variable is a random variable whose cumulative distribution function is continuous everywhere. There are no " gaps ", which would correspond to numbers which have a finite probability of occurring . Instead, continuous random variables almost never take an exact prescribed value c (formally, ∀ c ∈ R : Pr ( X = c ) = 0 {\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0} ) but there

1445-402: A direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, X = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any range of values. For example, the probability of choosing

1530-469: A housing development in Fourth Ward, Houston Apple Valley Airport (California) , from its IATA airport code Approach Procedure with Vertical guidance, a type of Instrument approach in aviation APV (NMDAR antagonist) , or AP5, a selective NMDA receptor antagonist APV plc , a former company making process equipment Asia Pacific Vision , a television content provider Chevrolet Lumina APV ,

1615-512: A life aged x {\displaystyle x} . The actuarial present value of one unit of an n -year term insurance policy payable at the moment of death can be found similarly by integrating from 0 to n . The actuarial present value of an n year pure endowment insurance benefit of 1 payable after n years if alive, can be found as In practice the information available about the random variable G (and in turn T ) may be drawn from life tables, which give figures by year. For example,

1700-401: A minivan manufactured and marketed by General Motors Suzuki APV , a microvan manufactured and marketed by Suzuki Amazon Prime Video Armored protected vehicle, a kind of armoured fighting vehicle A US Navy hull classification symbol: Transport and aircraft ferry (APV) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1785-462: A number in [0, 180] is 1 ⁄ 2 . Instead of speaking of a probability mass function, we say that the probability density of X is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set. More formally, given any interval I = [

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1870-519: A particular such sigma-algebra is used, the Borel σ-algebra , which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals. The measure-theoretic definition is as follows. Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be

1955-538: A pre-determined benefit either at or soon after the insured's death. The symbol (x) is used to denote "a life aged x " where x is a non-random parameter that is assumed to be greater than zero. The actuarial present value of one unit of whole life insurance issued to (x) is denoted by the symbol A x {\displaystyle \,A_{x}} or A ¯ x {\displaystyle \,{\overline {A}}_{x}} in actuarial notation . Let G>0 (the "age at death") be

2040-403: A probability distribution, if X {\displaystyle X} is real-valued, can always be captured by its cumulative distribution function and sometimes also using a probability density function , f X {\displaystyle f_{X}} . In measure-theoretic terms, we use the random variable X {\displaystyle X} to "push-forward"

2125-471: A random variable X {\displaystyle X} on Ω {\displaystyle \Omega } and a Borel measurable function g : R → R {\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} } , then Y = g ( X ) {\displaystyle Y=g(X)} is also a random variable on Ω {\displaystyle \Omega } , since

2210-405: A random variable X {\displaystyle X} yields the probability distribution of X {\displaystyle X} . The probability distribution "forgets" about the particular probability space used to define X {\displaystyle X} and only records the probabilities of various output values of X {\displaystyle X} . Such

2295-495: A random variable involves measure theory . Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox ) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally,

2380-484: A sample space Ω {\displaystyle \Omega } as a set of possible outcomes to a measurable space E {\displaystyle E} . The technical axiomatic definition requires the sample space Ω {\displaystyle \Omega } to be a sample space of a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )} (see

2465-445: A singular part. An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping

2550-417: A three year term life insurance of $ 100,000 payable at the end of year of death has actuarial present value For example, suppose that there is a 90% chance of an individual surviving any given year (i.e. T has a geometric distribution with parameter p = 0.9 and the set {1, 2, 3, ...} for its support). Then and at interest rate 6% the actuarial present value of one unit of the three year term insurance

2635-459: A whole life insurance benefit of 1 payable at time T . Then: where i is the effective annual interest rate and δ is the equivalent force of interest . To determine the actuarial present value of the benefit we need to calculate the expected value E ( Z ) {\displaystyle \,E(Z)} of this random variable Z . Suppose the death benefit is payable at the end of year of death. Then T(G, x) := ceiling (G - x)

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2720-617: Is F {\displaystyle {\mathcal {F}}} -measurable; X − 1 ( B ) ∈ F {\displaystyle X^{-1}(B)\in {\mathcal {F}}} , where X − 1 ( B ) = { ω : X ( ω ) ∈ B } {\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}} . This definition enables us to measure any subset B ∈ E {\displaystyle B\in {\mathcal {E}}} in

2805-599: Is g {\displaystyle g} 's inverse function ) and is either increasing or decreasing , then the previous relation can be extended to obtain With the same hypotheses of invertibility of g {\displaystyle g} , assuming also differentiability , the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to y {\displaystyle y} , in order to obtain If there

2890-411: Is so the actuarial present value of the $ 100,000 insurance is $ 24,244.85. In practice the benefit may be payable at the end of a shorter period than a year, which requires an adjustment of the formula. The actuarial present value of a life annuity of 1 per year paid continuously can be found in two ways: Aggregate payment technique (taking the expected value of the total present value ): This

2975-453: Is proportional to the length of the subinterval, that is, if a ≤ c ≤ d ≤ b , one has Pr ( X I ∈ [ c , d ] ) = d − c b − a {\displaystyle \Pr \left(X_{I}\in [c,d]\right)={\frac {d-c}{b-a}}} where the last equality results from the unitarity axiom of probability. The probability density function of

3060-414: Is real-valued , i.e. E = R {\displaystyle E=\mathbb {R} } . In some contexts, the term random element (see extensions ) is used to denote a random variable not of this form. When the image (or range) of X {\displaystyle X} is finitely or infinitely countable , the random variable is called a discrete random variable and its distribution

3145-449: Is a discrete probability distribution , i.e. can be described by a probability mass function that assigns a probability to each value in the image of X {\displaystyle X} . If the image is uncountably infinite (usually an interval ) then X {\displaystyle X} is called a continuous random variable . In the special case that it is absolutely continuous , its distribution can be described by

3230-405: Is a positive probability that its value will lie in particular intervals which can be arbitrarily small . Continuous random variables usually admit probability density functions (PDF), which characterize their CDF and probability measures ; such distributions are also called absolutely continuous ; but some continuous distributions are singular , or mixes of an absolutely continuous part and

3315-494: Is a real-valued random variable if This definition is a special case of the above because the set { ( − ∞ , r ] : r ∈ R } {\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}} generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using

3400-406: Is called a " continuous uniform random variable" (CURV) if the probability that it takes a value in a subinterval depends only on the length of the subinterval. This implies that the probability of X I {\displaystyle X_{I}} falling in any subinterval [ c , d ] ⊆ [ a , b ] {\displaystyle [c,d]\subseteq [a,b]}

3485-407: Is common to consider the special cases of discrete random variables and absolutely continuous random variables , corresponding to whether a random variable is valued in a countable subset or in an interval of real numbers . There are other important possibilities, especially in the theory of stochastic processes , wherein it is natural to consider random sequences or random functions . Sometimes

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3570-528: Is it that the value of X {\displaystyle X} is equal to 2?". This is the same as the probability of the event { ω : X ( ω ) = 2 } {\displaystyle \{\omega :X(\omega )=2\}\,\!} which is often written as P ( X = 2 ) {\displaystyle P(X=2)\,\!} or p X ( 2 ) {\displaystyle p_{X}(2)} for short. Recording all these probabilities of outputs of

3655-418: Is no invertibility of g {\displaystyle g} but each y {\displaystyle y} admits at most a countable number of roots (i.e., a finite, or countably infinite, number of x i {\displaystyle x_{i}} such that y = g ( x i ) {\displaystyle y=g(x_{i})} ) then the previous relation between

3740-554: Is not equal to f ( E ⁡ [ X ] ) {\displaystyle f(\operatorname {E} [X])} . Once the "average value" is known, one could then ask how far from this average value the values of X {\displaystyle X} typically are, a question that is answered by the variance and standard deviation of a random variable. E ⁡ [ X ] {\displaystyle \operatorname {E} [X]} can be viewed intuitively as an average obtained from an infinite population,

3825-396: Is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted E ⁡ [ X ] {\displaystyle \operatorname {E} [X]} , and also called the first moment . In general, E ⁡ [ f ( X ) ] {\displaystyle \operatorname {E} [f(X)]}

3910-399: Is similar to the method for a life insurance policy. This time the random variable Y is the total present value random variable of an annuity of 1 per year, issued to a life aged x , paid continuously as long as the person is alive, and is given by: where T=T(x) is the future lifetime random variable for a person age x . The expected value of Y is: Current payment technique (taking

3995-471: Is the Lebesgue measure in the case of continuous random variables, or the counting measure in the case of discrete random variables). The underlying probability space Ω {\displaystyle \Omega } is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on

4080-442: Is the number of "whole years" (rounded upwards) lived by (x) beyond age x , so that the actuarial present value of one unit of insurance is given by: where t p x {\displaystyle {}_{t}p_{x}} is the probability that (x) survives to age x+t , and q x + t {\displaystyle \,q_{x+t}} is the probability that (x+t) dies within one year. If

4165-415: Is the uniform distribution on the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Samples of any desired probability distribution D {\displaystyle \operatorname {D} } can be generated by calculating the quantile function of D {\displaystyle \operatorname {D} } on a randomly-generated number distributed uniformly on

4250-409: The ( E , E ) {\displaystyle (E,{\mathcal {E}})} -valued random variable is called an E {\displaystyle E} -valued random variable . Moreover, when the space E {\displaystyle E} is the real line R {\displaystyle \mathbb {R} } , then such a real-valued random variable is called simply

4335-486: The Iverson bracket , and has the value 1 if X {\displaystyle X} has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined. A new random variable Y can be defined by applying a real Borel measurable function g : R → R {\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} } to

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4420-424: The distribution of the random variable X {\displaystyle X} . Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function f ( X ) = X {\displaystyle f(X)=X} of

4505-402: The interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory , a random variable is defined as a measurable function from

4590-440: The measure-theoretic definition ). A random variable is often denoted by capital Roman letters such as X , Y , Z , T {\displaystyle X,Y,Z,T} . The probability that X {\displaystyle X} takes on a value in a measurable set S ⊆ E {\displaystyle S\subseteq E} is written as In many cases, X {\displaystyle X}

4675-423: The probability density functions can be generalized with where x i = g i − 1 ( y ) {\displaystyle x_{i}=g_{i}^{-1}(y)} , according to the inverse function theorem . The formulas for densities do not demand g {\displaystyle g} to be increasing. In the measure-theoretic, axiomatic approach to probability, if

4760-406: The random variable that models the age at which an individual, such as (x) , will die. And let T (the future lifetime random variable) be the time elapsed between age- x and whatever age (x) is at the time the benefit is paid (even though (x) is most likely dead at that time). Since T is a function of G and x we will write T=T(G,x) . Finally, let Z be the present value random variable of

4845-448: The sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from

4930-546: The "law of X {\displaystyle X} ". The density f X = d p X / d μ {\displaystyle f_{X}=dp_{X}/d\mu } , the Radon–Nikodym derivative of p X {\displaystyle p_{X}} with respect to some reference measure μ {\displaystyle \mu } on R {\displaystyle \mathbb {R} } (often, this reference measure

5015-404: The article on quantile functions for fuller development. Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their height. Associated with the random variable is a probability distribution that allows the computation of the probability that

5100-698: The benefit is payable at the moment of death, then T(G,x): = G - x and the actuarial present value of one unit of whole life insurance is calculated as where f T {\displaystyle f_{T}} is the probability density function of T , t p x {\displaystyle \,_{t}p_{x}} is the probability of a life age x {\displaystyle x} surviving to age x + t {\displaystyle x+t} and μ x + t {\displaystyle \mu _{x+t}} denotes force of mortality at time x + t {\displaystyle x+t} for

5185-400: The case where the annuity and life assurance are not whole life, one should replace the assurance with an n-year endowment assurance (which can be expressed as the sum of an n-year term assurance and an n-year pure endowment), and the annuity with an n-year annuity due. Random variable A random variable (also called random quantity , aleatory variable , or stochastic variable ) is

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5270-550: The coin is a fair coin , Y has a probability mass function f Y {\displaystyle f_{Y}} given by: f Y ( y ) = { 1 2 , if  y = 1 , 1 2 , if  y = 0 , {\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\[6pt]{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}} A random variable can also be used to describe

5355-461: The composition of measurable functions is also measurable . (However, this is not necessarily true if g {\displaystyle g} is Lebesgue measurable . ) The same procedure that allowed one to go from a probability space ( Ω , P ) {\displaystyle (\Omega ,P)} to ( R , d F X ) {\displaystyle (\mathbb {R} ,dF_{X})} can be used to obtain

5440-414: The definition above is valid for any measurable space E {\displaystyle E} of values. Thus one can consider random elements of other sets E {\displaystyle E} , such as random Boolean values , categorical values , complex numbers , vectors , matrices , sequences , trees , sets , shapes , manifolds , and functions . One may then specifically refer to

5525-498: The dice are fair ) has a probability mass function f X given by: f X ( S ) = min ( S − 1 , 13 − S ) 36 ,  for  S ∈ { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } {\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}} Formally,

5610-419: The different random variables to covary ). For example: If a random variable X : Ω → R {\displaystyle X\colon \Omega \to \mathbb {R} } defined on the probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )} is given, we can ask questions like "How likely

5695-437: The fact that { ω : X ( ω ) ≤ r } = X − 1 ( ( − ∞ , r ] ) {\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])} . The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it

5780-416: The height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm. Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values –

5865-467: The longer the period, the smaller the present value is due to two effects: Conversely, for contracts costing an equal lumpsum and having the same internal rate of return , the longer the period between payments, the larger the total payment per year. The APV of whole-life assurance can be derived from the APV of a whole-life annuity-due this way: This is also commonly written as: In the continuous case, In

5950-431: The measure P {\displaystyle P} on Ω {\displaystyle \Omega } to a measure p X {\displaystyle p_{X}} on R {\displaystyle \mathbb {R} } . The measure p X {\displaystyle p_{X}} is called the "(probability) distribution of X {\displaystyle X} " or

6035-546: The members of which are particular evaluations of X {\displaystyle X} . Mathematically, this is known as the (generalised) problem of moments : for a given class of random variables X {\displaystyle X} , find a collection { f i } {\displaystyle \{f_{i}\}} of functions such that the expectation values E ⁡ [ f i ( X ) ] {\displaystyle \operatorname {E} [f_{i}(X)]} fully characterise

6120-498: The outcomes of a real-valued random variable X {\displaystyle X} . That is, Y = g ( X ) {\displaystyle Y=g(X)} . The cumulative distribution function of Y {\displaystyle Y} is then If function g {\displaystyle g} is invertible (i.e., h = g − 1 {\displaystyle h=g^{-1}} exists, where h {\displaystyle h}

6205-676: The probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum PMF ⁡ ( 0 ) + PMF ⁡ ( 2 ) + PMF ⁡ ( 4 ) + ⋯ {\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots } . In examples such as these,

6290-589: The process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers n 1 and n 2 from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable X given by the function that maps the pair to the sum: X ( ( n 1 , n 2 ) ) = n 1 + n 2 {\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}} and (if

6375-488: The random variable have a well-defined probability. When E {\displaystyle E} is a topological space , then the most common choice for the σ-algebra E {\displaystyle {\mathcal {E}}} is the Borel σ-algebra B ( E ) {\displaystyle {\mathcal {B}}(E)} , which is the σ-algebra generated by the collection of all open sets in E {\displaystyle E} . In such case

6460-403: The random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the real-valued function [ X = green ] {\displaystyle [X={\text{green}}]} can be constructed; this uses

6545-443: The random variation of non-numerical data structures . In some cases, it is nonetheless convenient to represent each element of E {\displaystyle E} , using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space Ω {\displaystyle \Omega } , which allows

6630-551: The same random person, for example so that questions of whether such random variables are correlated or not can be posed. If { a n } , { b n } {\textstyle \{a_{n}\},\{b_{n}\}} are countable sets of real numbers, b n > 0 {\textstyle b_{n}>0} and ∑ n b n = 1 {\textstyle \sum _{n}b_{n}=1} , then F = ∑ n b n δ

6715-452: The set of values that the random variable can take (such as the set of real numbers), and a member of E {\displaystyle {\mathcal {E}}} is a "well-behaved" (measurable) subset of E {\displaystyle E} (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for

6800-495: The target space by looking at its preimage, which by assumption is measurable. In more intuitive terms, a member of Ω {\displaystyle \Omega } is a possible outcome, a member of F {\displaystyle {\mathcal {F}}} is a measurable subset of possible outcomes, the function P {\displaystyle P} gives the probability of each such measurable subset, E {\displaystyle E} represents

6885-476: The title APV . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=APV&oldid=1187872358 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Actuarial present value Whole life insurance pays

6970-435: The total present value of the function of time representing the expected values of payments): where F ( t ) is the cumulative distribution function of the random variable T . The equivalence follows also from integration by parts. In practice life annuities are not paid continuously. If the payments are made at the end of each period the actuarial present value is given by Keeping the total payment per year equal to 1,

7055-456: The unit interval. This exploits properties of cumulative distribution functions , which are a unifying framework for all random variables. A mixed random variable is a random variable whose cumulative distribution function is neither discrete nor everywhere-continuous . It can be realized as a mixture of a discrete random variable and a continuous random variable; in which case the CDF will be

7140-461: The value −1. Other ranges of values would have half the probabilities of the last example. Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see Lebesgue's decomposition theorem § Refinement . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers). The most formal, axiomatic definition of

7225-469: The weighted average of the CDFs of the component variables. An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of 1 ⁄ 2 that this random variable will have

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