c) Consider the following random experiment. Suppose you roll a fair 4-sided die

Question

c) Consider the following random experiment. Suppose you roll a fair 4-sided die, and let N be the outcome. Let X be a continuous uniform RV such that X ~ U(0,/ ). For example, if you roll the die and get N 2, then X~U(0,2). What is P(X> 0.5). Hint: Use law of total probability, where the possibilities for N form the partition. d) Consider the random experiment from c). Given that X 0.5, what is the probability that you rolled a 4 on the die? Hint: Use Bayes Rule, and I am asking for P(N-41X 0.5)

Explanation / Answer

c) P(X>0.5)=P(1 apperas and x>0.5 +2 apperas and x>0.5 +3 apperas and x>0.5 +4 apperas and x>0.5 )

=(1/4)*(1-0.5)/(1-0)+(1/4)*(2-0.5)/(2-0)+(1/4)*(3-0.5)/(3-0)+(1/4)*(4-0.5)/(4-0) =0.7396

d) probability P(X<0.5)=1-P(X>0.5)=1-0.7396=0.2604

hence P(N=4|X<0.5)=P(N=4 and X<0.5)/P(X<0.05)= (1/4)*((0.5-0)/(4-0))/0.2604=0.1200

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