# Probability and Statistics

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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