Exercise 2. Let p = [0, 1] and let random variables X₁,..., Xn be i.i.d., such that P(X₂ = 1) = p and P(X₂ = 0) = 1-p for all i = {1, 2,...,n}. Finally, let X be the average of the X₁. a) Determine E(X;) and Var(X₂). b) Determine E(X) and Var(X). c) Show that P(X = k/n) = (*)p² (1 − p)”-k,
Exercise 2. Let p = [0, 1] and let random variables X₁,..., Xn be i.i.d., such that P(X₂ = 1) = p and P(X₂ = 0) = 1-p for all i = {1, 2,...,n}. Finally, let X be the average of the X₁. a) Determine E(X;) and Var(X₂). b) Determine E(X) and Var(X). c) Show that P(X = k/n) = (*)p² (1 − p)”-k,
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.7: More On Inequalities
Problem 44E
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![Exercise 2. Let p = [0, 1] and let random variables X₁,..., Xn be i.i.d., such that P(X; = 1) = p
and P(X₂ = 0) = 1-p for all i = {1,2,...,n}. Finally, let X be the average of the X₁.
a) Determine E(X₂) and Var(X₂).
b) Determine E(X) and Var (X).
c) Show that
P(X = k/n) = (7) p² (1
- (^)^(1-2)^-^.
k p)"-
Exercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the
expectation E((X -a)²).
a) Write E((X-a)²) in terms of a, μ and o².
b) For which value a is E((X - a)2) minimal?
c) For the value a from part (b), what is E((X− a)²)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed029406-a1c1-473f-a3a0-6fd0fbd8e89d%2Fa35b7717-b77a-47c3-a7ac-2daabba75aca%2Fx4f80w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 2. Let p = [0, 1] and let random variables X₁,..., Xn be i.i.d., such that P(X; = 1) = p
and P(X₂ = 0) = 1-p for all i = {1,2,...,n}. Finally, let X be the average of the X₁.
a) Determine E(X₂) and Var(X₂).
b) Determine E(X) and Var (X).
c) Show that
P(X = k/n) = (7) p² (1
- (^)^(1-2)^-^.
k p)"-
Exercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the
expectation E((X -a)²).
a) Write E((X-a)²) in terms of a, μ and o².
b) For which value a is E((X - a)2) minimal?
c) For the value a from part (b), what is E((X− a)²)?
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