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Supplement_3 - DISCRETE RANDOM VARIABLES DISCRETE RANDOM...

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≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈ DISCRETE RANDOM VARIABLES ≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈ DISCRETE RANDOM VARIABLES Documents prepared for use in course C22.0103.001 , New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced here. The related concepts of mean, expected value, variance, and standard deviation are also discussed. Binomial random variable examples page 5 Here are a number of interesting problems related to the binomial distribution. Hypergeometric random variable page 9 Poisson random variable page 15 Covariance for discrete random variables page 19 This concept is used for general random variables, but here the arithmetic for the discrete case is illustrated. revised by Avi Giloni on September 1, 2005. Gary Simon, 2005
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≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈ DISCRETE RANDOM VARIABLES ≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈
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Discrete Random Variables and Related Properties {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ Discrete random variables are obtained by counting and have values for which there are no in-between values. These values are typically the integers 0, 1, 2, …. Random variables are usually denoted by upper case (capital) letters. The possible values are denoted by the corresponding lower case letters, so that we talk about events of the form [ X = x ]. The random variables are described by their probabilities. For example, consider random variable X with probabilities x 0 ±1234 ±5 ± P[ X = x ] 0.05 0.10 0.20 0.40 0.15 0.10 You can observe that the probabilities sum to 1. The notation P ( x ) is often used for P[ X = x ]. The notation f ( x ) is also used. In this example, P (4) = 0.15. The symbol P (or f ) denotes the probability function, also called the probability mass function. The cumulative probabilities are given as . The interpretation is that F ( x ) is the probability that X will take a value less than or equal to x . The function F is called the cumulative distribution function (CDF). This is the only notation that is commonly used. For our example, Fx Pi ix () = F (3) = P[ X 3] = P[ X =0] + P[ X =1] + P[ X =2] + P[ X =3] = 0.05 + 0.10 + 0.20 + 0.40 = 0.75 One can of course list all the values of the CDF easily by taking cumulative sums: x 0 ± P[ X = x ] 0.05 0.10 0.20 0.40 0.15 0.10 F ( x ) 0.05 0.15 0.35 0.75 0.90 1.00 The values of F increase. The expected value of X is denote either as E( X ) or as µ . It’s defined as . The calculation for this example is EP ( P ) [ Xx xx X == ∑∑ ] x = E( X ) = 0 × 0.05 + 1 × 0.10 + 2 × 0.20 + 3 × 0.40 + 4 × 0.15 + 5 × 0.10 = 0.00 + 0.10 + 0.40 + 1.20 + 0.60 + 0.50 = 2.80 This is also said to be the mean of the probability distribution of X . { page gs2003 3
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Discrete Random Variables and Related Properties {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ The probability distribution of X also has a standard deviation, but one usually first defines the variance. The variance of X , denoted as Var( X ) or σ 2 , or perhaps , is 2 X σ Var( X ) = () ( 2 P x ) x x −µ = ( 2 P x ) x Xx µ= This is the expected square of the difference between X and its expected value, µ . We can calculate this for our example: x 01234 ±5 x - 2.8 -2.8 -1.8 -0.8 0.2 1.2 2.2 ( x - 2.8) 2 7.84 3.24 0.64 0.04 1.44 4.84 P[ X = x ] 0.05 0.10 0.20 0.40 0.15 0.10 ( x -2.8) 2 P[ X = x ] 0.392 0.324 0.128 0.016 0.216 0.484 The variance is the sum of the final row. This value is 1.560.
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This note was uploaded on 05/24/2008 for the course ACC 203 taught by Professor Choi during the Spring '08 term at NYU.

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Supplement_3 - DISCRETE RANDOM VARIABLES DISCRETE RANDOM...

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