Problems 1. Let A and B be events with P(A) #0 and P(B) 0. If P(A/B)...

 

Transcribed Image Text:

Problems 1. Let A and B be events with P(A) #0 and P(B) 0. If P(A/B) > P(A), prove that P(BA) > P(B). 2. Suppose that we throw two unbiased dice independently. Let A = { sum of the faces = 8} and let B = { faces are equal}. Compute P(BIA). 3. A jar contains w white balls, b black balls and r red balls. Find the probability of a white ball being drawn before a black ball if each ball is replaced after being drawn. 4. Let X be a random variable with probability mass function f(x) = for x = 1,2,3,.... Using only the definition of the expected value, compute the expected value of X. Please note, you are not allowed to solve this problem by using the moment generating function. 5. Let X be a random variable with probability mass function f(x) = p¹q for x = 1,2,3.... where 0 < p < 1 and q=1-p. Prove that P(X>k+j|X > k) = P(X> j), where J and k are non-negative integers. Problems 1. Let A and B be events with P(A) #0 and P(B) 0. If P(A/B) > P(A), prove that P(BA) > P(B). 2. Suppose that we throw two unbiased dice independently. Let A = { sum of the faces = 8} and let B = { faces are equal}. Compute P(BIA). 3. A jar contains w white balls, b black balls and r red balls. Find the probability of a white ball being drawn before a black ball if each ball is replaced after being drawn. 4. Let X be a random variable with probability mass function f(x) = for x = 1,2,3,.... Using only the definition of the expected value, compute the expected value of X. Please note, you are not allowed to solve this problem by using the moment generating function. 5. Let X be a random variable with probability mass function f(x) = p¹q for x = 1,2,3.... where 0 < p < 1 and q=1-p. Prove that P(X>k+j|X > k) = P(X> j), where J and k are non-negative integers.

Answer rating: 100% (QA)

1We have PAB PA which implies that PABPB PA and hence PAB PAPB Also since PB 0 we can multiply both sides of the inequality by PB to get PBA PAPB 0 an... View the full answer