Probability and Statistics

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

proof Let X˜N(μ,σ2) be a normal random variable. Derive F_X(x)\,, the cdf of X\,, in terms of the error function from calculus, {\rm Erf}(x)=\int_0^xe^{-y^2}\,dy.

solution A continuous random variable X\, has a pdf of the form f(x)=\begin{cases}ce^{-2x}&x>1\\0&x\leq1,\end{cases} where c is a constant. Find the value of c and the cdf of X\,.

solution Let us have an array of N numbered elements, p.e., \{1,2,\ldots,N\}. Let the array be randomly reordenated. Find the probability of finding none of the elements in its original position when N\to\infty.

Wind Prediction for "Wind Prediction and Monte Carlo Simulation"

solution Calculate the joint probability distribution \int_1^2\int_2^3 6e^{-2x_1-3x_2}\ dx_1\ dx_2

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