Question

Many random processes are well understood. Let’s study two of them. A Bernoulli process is a random process with only two possible outcomes: “Head” or “Tail”, “Success” or “Failure”, 1 or 0, etc. Examples: ipping a coin; winning the grand prize in a lottery; whether it rains on any given day. Let Y be the random variable of a Bernoulli process. It is customary to dene the sample space as S = {1,0}, where 1 denotes “Head” or “Success” and 0 denotes “Tail” or “Failure”.
If P(Y = 1) = p, where p is some real number between 0 and 1, then clearly P(Y = 0) = 1−p. Think of p as the “success rate”, or the probability of getting a “Head” in a single coin ip. We call p the parameter of the Bernoulli process, and write: Y ∼ Bernoulli(p) to mean that Y is a random variable of a Bernoulli process with success rate p. Example: ipping a fair coin once is Bernoulli process with success rate p = 0.5.
Another common random process is the Binomial process. This may be dened as the sum of n independent Bernoulli processes. Consider ipping a coin n times, where each ip is the random variable Yi ∼ Bernoulli(p), for i = 1,2,...,n. Now let X = Y1 + Y2 + ...Yn. Then the sample space is U = {0,1,2,...,n}, and the random variable X can take any value in U. You can interpret X as the number of “Heads” when the coin is ipped n times.
The probability of getting k “Heads” in n ips is given by: P(X = k) =n kpk(1−p)n−k, for k = 0,1,2,...,n. And we say: X ∼ Binom(n,p) to mean that X follows a Binomial process with parameters n and p. Note that a Binomial process has two parameters.
(a) (1 mark) If Y ∼ Bernoulli(p), then show that E[Y ] = p. (b) (1 mark) If X ∼ Binom(n,p), then show that E[X] = np. (c) You own a restaurant, and from experience, you know that 3 out of every 5 customers will ask for ice water with their meal. A sports team of 12 people has booked your restaurant on Sunday to celebrate their victory. Model this problem as follows:
Let X be the number of people who want ice water with their meal. Then X ∼ Binom(12,p), where p = 3/5. i. Calculate the probability that at least 3 people will ask for ice water. ii. (1 mark) You would like to prepare enough ice water for Sunday’s party. What is the expected number of people who would want ice water?

Answer 1

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