(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).
(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).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
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).
(b) Now, suppose we toss n independent coins, each of which has probability p of turning up heads. Let N
denote 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). Such
a random variable is said to be Binomially distributed with n trials and probability of success p. We write
X 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 of
tosses. What is P(N = n) for n € N? Likewise, what is E[N]? Var[N]? The random variable with said
distribution is said to be Geometrically distributed with parameter p. We write this as N ~ Geo (p).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1ceeb10-3a2e-403a-927f-8b66ea513861%2Fffa27217-55eb-4f5c-8f49-b9eddb30e56c%2Fyvwtzri_processed.png&w=3840&q=75)
Transcribed Image Text:(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).
(b) Now, suppose we toss n independent coins, each of which has probability p of turning up heads. Let N
denote 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). Such
a random variable is said to be Binomially distributed with n trials and probability of success p. We write
X 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 of
tosses. What is P(N = n) for n € N? Likewise, what is E[N]? Var[N]? The random variable with said
distribution is said to be Geometrically distributed with parameter p. We write this as N ~ Geo (p).
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