Denote by N the outcome of rolling a six-faced fair die. When the outcome N is...

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Denote by N the outcome of rolling a six-faced fair die. When the outcome N is known to us, we take a random sample of size N from the normal distribution N(10, 22). Denote the random sample by {X1,..., XN}, where X and X; are independent whenever i j. After the sample has been taken, we write down these values on separate cards so that each card has one and only one value. If the value on a card is greater than 8.5, the card will be stamped. Let Y denote the number of stamped cards. (a) Find Pr(X > 8.5). (b) Find Pr(Y 5|N = 6). (2 marks) (4 marks) (c) Find Pr(Y 5). (4 marks) (d) Find Pr(N = 6|Y 5). (4 marks) (e) Are the events (Y 5} and {N = 6} independent? Justify your answer. (2 marks) (f) Let X denote the smallest number among X,..., XN, where N itself is a random variable. Find Pr(X > 8.5). (6 marks) Denote by N the outcome of rolling a six-faced fair die. When the outcome N is known to us, we take a random sample of size N from the normal distribution N(10, 22). Denote the random sample by {X1,..., XN}, where X and X; are independent whenever i j. After the sample has been taken, we write down these values on separate cards so that each card has one and only one value. If the value on a card is greater than 8.5, the card will be stamped. Let Y denote the number of stamped cards. (a) Find Pr(X > 8.5). (b) Find Pr(Y 5|N = 6). (2 marks) (4 marks) (c) Find Pr(Y 5). (4 marks) (d) Find Pr(N = 6|Y 5). (4 marks) (e) Are the events (Y 5} and {N = 6} independent? Justify your answer. (2 marks) (f) Let X denote the smallest number among X,..., XN, where N itself is a random variable. Find Pr(X > 8.5). (6 marks)