# Probability and Statistics

### From Exampleproblems

Solutions Manual for "Statistical Inference by Berger and Casella"

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining one pair.

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining two pairs.

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining three of a kind.

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining a straight (but not a straight flush).

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining a flush (but not a straight flush).

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining a full house.

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining four of a kind.

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining a straight flush (but not a royal flush).

solution Five cards are randomly dealt from a standard deck of cards. Find the probability of obtaining a royal flush.

solution In Morse code, dots and dashes are sent in the ratio 3:4, but transmission errors cause a dot to be come a dash with probability 1/4 while a dash is incorrectly read as a dot with probability 1/6. Assuming any two signals are independent of each other, find the probability distribution of the message actually sent when a "dash-dot-dot" is received.

proof Let be the number of events per unit time, so that *X*˜Poisson(λ). Prove that , the time between successive events, has an exponential distribution with rate parameter .

proof Let *X*˜N(μ,σ^{2}) be a normal random variable. Derive , the cdf of , in terms of the error function from calculus, .

solution A continuous random variable has a pdf of the form
where *c* is a constant. Find the value of *c* and the cdf of .

solution Let us have an array of *N* numbered elements, p.e., . Let the array be randomly reordenated. Find the probability of finding none of the elements in its original position when .

Wind Prediction for "Wind Prediction and Monte Carlo Simulation"

solution Calculate the joint probability distribution