# 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 $X\sim {{\rm {Poisson}}}(\lambda )$. Prove that $T\,$, the time between successive events, has an exponential distribution with rate parameter $\lambda \,$.

proof Let $X\sim {{\rm {N}}}(\mu ,\sigma ^{2})$ be a normal random variable. Derive $F_{X}(x)\,$, the cdf of $X\,$, in terms of the error function from calculus, ${{\rm {Erf}}}(x)=\int _{0}^{x}e^{{-y^{2}}}\,dy$.

solution A continuous random variable $X\,$ has a pdf of the form $f(x)={\begin{cases}ce^{{-2x}}&x>1\\0&x\leq 1,\end{cases}}$ 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 $\int _{1}^{2}\int _{2}^{3}6e^{{-2x_{1}-3x_{2}}}\ dx_{1}\ dx_{2}$