Question

The probability of getting a full house in a poker hand is 6/6145. Let, X be the random variable that counts the number of full houses in a sequence of 500 poker hands; each poker hand being dealt after the previous hand is returned to the deck and the deck is reshuffled.

What type of random variable is X?
(b) The random variable X can be approximated by a Poisson random variable Y
since the probability of getting a fun house is small and the number of hands
(namely, 500) is large. Write down the pdf of the Poisson random variable Y
that is an approximation to X.

Answer 1

X, the random variable counting the number of full houses in a sequence of 500 poker hands, is a binomial random variable, but it can be approximated by a Poisson random variable due to the small probability of success and large number of hands. The pdf of the Poisson random variable is given by P(Y = y) = (e^(-λ) * λ^y) / y! where λ is the expected value.

Step-by-step explanation:

X, the random variable that counts the number of full houses in a sequence of 500 poker hands, is a binomial random variable. A binomial random variable represents the number of successes in a series of independent trials, where each trial has the same probability of success. In this case, a full house is the success, and the probability of getting a full house in one poker hand is 6/6145.

However, since the probability of getting a full house is small (6/6145) and the number of hands (500) is large, X can be approximated by a Poisson random variable, denoted as Y. The pdf of a Poisson random variable is given by the formula P(Y = y) = (e^(-λ) * λ^y) / y!, where λ is the expected value of Y.