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1. Roll an even dice and observe the number N on the uppermost face. Then toss...

1. Roll an even dice and observe the number N on the uppermost face. Then toss a fair coin N times and observe X, the total number of heads that appear in N tosses. (i) Write down the conditional probability mass function pX|N(·|3). (ii) What is P(X = 5)? (iii) What is E(X)?

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a! a(a - 1)(a 2)...1 a! ы(а - b)! binomial distributed variable X with sample size(n) and probability of success (p) For a P(

On a dice, there are 6 outcomes 1,2, 3, 4, 5,6} On an even dice, all the 6 outcomes are equally likely So, probability of any

An even dice is rolled and then. a fair coin is tossed the same number of times as the outcome on the dice The outcome on the

7 binomial distributed For n tosses of a fair coin, number of heads y are a 1 variable with probability of success(p S 2 So,

For N3, there will be 3 coin tosses Probability of observing 0 heads 3! (3 0)! 8 - () )* Probability of observing 1 head _ 3!

So, conditional probability mass function of X given N = 0 1 PxIN( 3) = 2 = 3 coloo co loo

b) The number of heads can be 5 when the coin is tossed at least 5 times Coin can be tossed at least 5 times when N 5 or N 6

P(X 5) P(X 5 and N 5)P(X5 and N 6) 5)*P(N 5)+P(X 5 N 6)*P(N 6) P(X 5 N 1 3 1 1 32 6 6 48 CO

c) For y tosses of a fair coin, expected number of heads: 1 y E[X | N yy* 2 2 n-6 E[X|N = n]- E[X] = P(N n) n=l n-6 1 EXIN 6

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