Ch2 Random Variables and Probability Distributions

# Ch2 Random Variables and Probability Distributions -...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter II Random Variables and Probability Distributions § 2.1 Random Variables Definition A random variable is a numerical valued function defined on a sample Ω : X space. In other words, a number ( ) ω X , providing a measure of the characteristic of interest, is assigned to each outcome in the sample space. Remark Always keep in mind that X is a function rather than a number. The value of X depends on the outcome. We write x X = to represent the event () { | x X = Ω } and x X to represent the event { } | x X Ω . Example 2.1 Let X be the number of aces in a hand of three cards drawn randomly from a deck of 52 cards. Denote A as an ace card and N as a non-ace card. Then Ω = {AAA, AAN, ANA, ANN, NAA, NAN, NNA, NNN} The space of X is {0, 1, 2, 3}. Hence X is discrete. { 3 , 2 , 1 , 0 : Ω X } such that 3 = AAA X ( ) 2 = = = NAA X ANA X AAN X ( ) 1 = = = NNA X NAN X ANN X 0 = NNN X Refer to the same example from Chapter I, we have { } ( ) 78262 . 0 0 = = = NNN P X P { } ( ) 20415 . 0 3 06805 . 0 , , 1 = × = = = NNA NAN ANN P X P { } ( ) 01302 . 0 3 00434 . 0 , , 2 = × = = = NAA ANA AAN P X P { } ( ) 00018 . 0 3 = = = AAA P X P p. 39

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Example 2.2 The annual income ω of a randomly selected citizen has a sample space [ ) = Ω , 0. Suppose the annual income is taxable if it exceeds c . Let X be the taxable income. Then the space of X is also [ n d ) , 0 a [ ) Ω , 0 : X such that () > = c c c X ωω , , 0 . Note : Conventionally, we use capital letters X , Y , … to denote random variables and small letters x , y , … the possible numerical values (or realizations ) of these variables. § 2.2 Distribution of the Discrete Type Definition A random variable X defined on the sample space Ω is called a discrete random variable if () () { } : Ω = Ω X X is countable (e.g. { } ,... 2 , 1 , 0 : Ω X ). Definition The probability mass function ( pmf ) of a discrete random variable X is defined as () ( ) x X P x p = = , ( ) Ω X x , where is the countable set of possible values of X . Ω X Example 2.3 For the previous example of card drawing, the pmf of X is 78262 . 0 0 = p , , 20415 . 0 1 = p ( ) 01302 . 0 2 = p , . 00018 . 0 3 = p p. 40
Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Conditions for a pmf Since pmf is defined through probability, we have the following conditions for p to be a valid pmf: 1. () 0 x for Ω p X x ; 0 = x p for Ω X x 2. 1 = Ω X x x p 3. ( ) = A x x p A X P where ( ) Ω X A Example 2.4 Is 6 x x p = , 3 , 2 , 1 = x a valid pmf ? ( ) x p 1/2 1/3 1/6 1 2 3 (){ } 3 , 2 , 1 = Ω X 1. 0 6 > = x x p for all 3 , 2 , 1 = x . 2. 1 2 1 3 1 6 1 3 1 = + + = = x x p 3. ( ) ( ) 2 1 2 = 1 2 + = p p X P p. 41

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Definition The ( cumulative ) distribution function ( cdf ) of the discrete random variable X is defined as () ( ) ( ) = = x t t p x X P x F , < < x . Example 2.5 Using previous example, () 6 x x p = , 3 , 2 , 1 = x , we have 6 1 1 1 1 = = = = X P X P F ( ) ( ) ( ) ... 99999 . 1 566 . 1 6 1 1 5 . 1 5 . 1 = = = = = = = F F X P X P F () ( ) () () 2 1 2 1 2 2 = + = = p p X P F () ( ) () ( ) ( ) 1 3 2 1 3 3 = + + = = p p p X P F As can be seen, the cdf of a discrete random variable would be a step-function with as the size of the jumps at the possible value x . x p F x 1
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Ch2 Random Variables and Probability Distributions -...

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