Exercise 26 Compute the expected values of random variables with Bernoulli(p), Binom(n,p), Geom(p) and NegBi- nom(k,p) distributions, respectively, in terms of the unknown parameters. Show your computations. Let X be a discrete random variable. (i) Prove that Var(X) = E(X²) –- (EX)². Hint: Use the definition of the variance, the binomial formula, and the facts you know about the expected value. Notice that EX is a real number and can be treated like a constant. (ii) Prove that Var(aX +b) = a²Var(X), for any a, be R.
Exercise 26 Compute the expected values of random variables with Bernoulli(p), Binom(n,p), Geom(p) and NegBi- nom(k,p) distributions, respectively, in terms of the unknown parameters. Show your computations. Let X be a discrete random variable. (i) Prove that Var(X) = E(X²) –- (EX)². Hint: Use the definition of the variance, the binomial formula, and the facts you know about the expected value. Notice that EX is a real number and can be treated like a constant. (ii) Prove that Var(aX +b) = a²Var(X), for any a, be R.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:Exercise 26 Compute the expected values of random variables with Bernoulli(p), Binom(n,p), Geom(p) and NegBi-
nom(k,p) distributions, respectively, in terms of the unknown parameters. Show your computations.
Let X be a discrete random variable.
(i) Prove that
Var(X) = E(X²) – (EX)².
Hint: Use the definition of the variance, the binomial formula, and the facts you know about the expected value.
Notice that EX is a real number and can be treated like a constant.
(ii) Prove that
Var(aX + b) = a²Var(X),
for any a, b e R.
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