a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {1, -a(1-2e-*₁) (1 - 2e-x²)]e*1-*2 ,0 ≤ x₁,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(k) = ( pqk-1, k = 1,2, lo, elsewhere i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K.
a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by f(x₁, x₂) = {1, -a(1-2e-*₁) (1 - 2e-x²)]e*1-*2 ,0 ≤ x₁,0 ≤ x₂. otherwise i) Find the marginal distribution of X₁. ii) Find E(X₁X₂). b) Consider a random variable K with parameter p, whose probability mass function (PMF) is given by f(k) = ( pqk-1, k = 1,2, lo, elsewhere i) Derive the moment generating function of K. ii) Use the result obtained in i), to find the expected value of K. iii) Use the result obtained in i), to find the variance of value of K.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 29CR
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![a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {[1-
-a(1-2e-*₁) (1 - 2e-x²)]e*1-*2
otherwise
,0 ≤ x₁,0 ≤ x₂.
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).
b) Consider a random variable K with parameter p, whose probability mass function (PMF) is
given by
f(k)
pqk-1, k=1,2,.
= lo, elsewhere
i) Derive the moment generating function of K.
ii) Use the result obtained in i), to find the expected value of K.
iii) Use the result obtained in i), to find the variance of value of K.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7f442cf-541d-4442-b6c8-0606c8499d0b%2F69fb2352-a611-410c-88cc-ee0cf49b852f%2Fgrtze8a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a) Suppose that, for -1 ≤ a ≤ 1, the probability density function of (X₁, X₂) is given by
f(x₁, x₂) = {[1-
-a(1-2e-*₁) (1 - 2e-x²)]e*1-*2
otherwise
,0 ≤ x₁,0 ≤ x₂.
i) Find the marginal distribution of X₁.
ii) Find E(X₁X₂).
b) Consider a random variable K with parameter p, whose probability mass function (PMF) is
given by
f(k)
pqk-1, k=1,2,.
= lo, elsewhere
i) Derive the moment generating function of K.
ii) Use the result obtained in i), to find the expected value of K.
iii) Use the result obtained in i), to find the variance of value of K.
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