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# 02_07 - E(X x f x x ⋅ f x 10 0.2 11 0.4 12 0.3 13 0.1 E(X...

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STAT 400 Examples for 02/07/2011 Spring 2011 A random variable associates a numerical value with each outcome of a random experiment. A random variable is said to be discrete if it has either a finite number of values or infinitely many values that can be arranged in a sequence. If a random variable represents some measurement on a continuous scale and therefore capable of assuming all values in an interval, it is called a continuous random variable. The probability distribution of a discrete random variable is a list of all its distinct numerical values along with their associated probabilities: x f ( x ) x 1 x 2 x 3 x n f ( x 1 ) f ( x 2 ) f ( x 3 ) f ( x n ) ! ! 1) for each x , 0 f ( x ) 1. 2) x f x all ) ( = 1 . 1.00 Often a formula can be used in place of a detailed list. 1. A balanced (fair) coin is tossed twice. Let x f ( x ) X denote the number of H's. Construct the probability distribution of X.

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2. Suppose a random variable X has the following probability distribution: x f ( x ) 10 0.20 11 0.40 12 0.30 13 0.10 a) Find the expected value of X,

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Unformatted text preview: E(X). x f ( x ) x ⋅ f ( x ) 10 0.2 11 0.4 12 0.3 13 0.1 E(X) = μ X = ∑ ⋅ x x f x all ) ( b) Find the variance of X, Var(X). x f ( x ) ( x-μ X ) ( x-μ X ) 2 ⋅ f ( x ) 10 0.2 11 0.4 12 0.3 13 0.1 Var(X) = 2 X σ = ( ) ∑ ⋅-x x f x all 2 X ) ( μ = E [ X -μ X ] 2 x f ( x ) x 2 ⋅ f ( x ) 10 0.2 11 0.4 12 0.3 13 0.1 Var(X) = 2 X σ = [ ] 2 all 2 E(X) ) ( x x f x-∑ ⋅ = E ( X 2 ) – [ E(X) ] 2 c) Find the standard deviation of X, SD(X). SD(X) = X σ = 2 X σ E ( g (X) ) = ∑ ⋅ x x f x g all ) ( ) ( E ( a ⋅ X + b ) = a ⋅ E ( X ). + b . Var ( a ⋅ X + b ) = a 2 ⋅ Var ( X ). SD ( a ⋅ X + b ) = | a | ⋅ SD ( X ). 3. Suppose E(X) = 7, SD(X) = 3. a) Y = 2 X + 3. Find E(Y) and SD(Y). b) W = 5 – 2 X. Find E(W) and SD(W). 4. Suppose a discrete random variable X has the following probability distribution: P( X = 0 ) = e 2-, P( X = k ) = ! 2 1 k k ⋅ , k = 1, 2, 3, … a) Find E ( X ). b) Find Var ( X )....
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02_07 - E(X x f x x ⋅ f x 10 0.2 11 0.4 12 0.3 13 0.1 E(X...

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