(a) Suppose we toss one biased coin with probability of heads p € (0, 1). Let X = 1 if the coin turns up heads,X = 0 if the coin lands tails. What is E[X]? What about Var[X]? X is said to be a Bernoulli randomvariable with probability of success p. We write X~ Ber(p).(b) Now, suppose we toss n independent coins, each of which has probability p of turning up heads. Let Ndenote the number of heads out of the n tosses. What is P(N = k), for k = 0, 1,..., n? What is E[N]?What about Var [N]? (Hint: these last two points can be solved easily with the previous question). Sucha random variable is said to be Binomially distributed with n trials and probability of success p. We writeX Bin(n, p) correspondingly.(c) Instead of tossing a fixed number of coins, you now toss independent coins until you see the first heads,where p is still the probability that any given coin lands heads up. Let N represent this number oftosses. What is P(N = n) for n € N? Likewise, what is E[N]? Var[N]? The random variable with saiddistribution is said to be Geometrically distributed with parameter p. We write this as N ~ Geo (p).

a) Suppose we toss one biased a coin with probability of head p.Now X01P(X=x)(1-p)p

Answered: (a) Suppose we toss one biased coin…

A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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(a) Suppose we toss one biased coin with probability of heads p € (0, 1). Let X = 1 if the coin turns up heads, X = 0 if the coin lands tails. What is E[X]? What about Var [X]? X is said to be a Bernoulli random variable with probability of success p. We write X~ Ber(p).