91影视

Let $X$be a Poisson random variable with parameter $位$. Show that $PX=i$increases monotonically and then decreases monotonically as$i$increases, reaching its maximum when $i$is the largest integer not exceeding $位$.

Hint: Consider $PX=i/PX=i鈭�1$.

A total of $2n$people, consisting of $n$married couples, are randomly divided into $n$pairs. Arbitrarily number the women, and let $W_i$denote the event that woman $i$is paired with her husband.

1. Find$P\left(Wi\right).$
2. For $i鈮�j,$find role="math" localid="1646662043709" $P\left(W_i鈭�W_j\right)$.
3. When $n$is large, approximate the probability that no wife is paired with her husband.
4. If each pairing must consist of a man and a woman, what does the problem reduce to?

Let X be a Poisson random variable with parameter 位.

• (a) Show that$P\left\{X\text{is even}\right\}=\frac{1}{2}\left[1+e^{鈭�2位}\right]$by using the result of Theoretical Exercise 4.15 and the relationship between Poisson and binomial random variables.
• (b) Verify the formula in part (a) directly by making use of the expansion of$e^{鈭�位}+e^{位}$

Suppose that the distribution function of X given by

$F\left(b\right)=\left\{\begin{array}{l}0鈥呪赌呪赌呪赌�b<0\\ \frac{b}{4}鈥呪赌呪赌呪赌�0鈮�b<1\\ \frac{1}{2}+\frac{b鈭�1}{4}鈥呪赌呪赌呪赌�1鈮�b<2\\ \frac{11}{12}鈥呪赌呪赌呪赌�2鈮�b<3\\ 1鈥呪赌呪赌呪赌�3鈮�b\end{array}\right.$

(a) Find $P\{X=i\},i=1,2,3$.

(b) Find $P\left\{\frac{1}{2}<X<\frac{3}{2}\right\}$.

Four independent flips of a fair coin are made. Let $X$ denote the number of heads obtained. Plot the probability mass function of the random variable $X-2$.

If the distribution function of $X$is given by

$F\left(b\right)=\left\{\begin{array}{l}0鈥呪赌呪赌呪赌�b<0\\ \frac{1}{2}鈥呪赌呪赌呪赌�0鈮�b<1\\ \frac{3}{5}鈥呪赌呪赌呪赌�1鈮�b<2\\ \frac{4}{5}鈥呪赌呪赌呪赌�2鈮�b<3\\ \frac{9}{10}鈥呪赌呪赌呪赌�3鈮�b<3.5\\ 1鈥呪赌呪赌呪赌�b鈮�3.5\end{array}\right.$

calculate the probability mass function of $X$.

When three friends go for coffee, they decide who will pay the check by each flipping a coin and then letting the 鈥渙dd person鈥� pay. If all three flips produce the same result (so that there is no odd person), then they make a second round of flips, and they continue to do so until there is an odd person. What is the probability that

1. exactly $3$rounds of flips are made?
2. more than $4$rounds are needed?

Show that $X$is a Poisson random variable with parameter $位$, then

$E\left[X^n\right]=位E\left[(X+1{)}^{n-1}\right]$

Now use this result to compute $E\left[X^3\right]$.

Suppose that $X$takes on one of the values$0,1$and$2$. If for some constant$c,P\{X=i\}=cP\{X=i-1\},i=1,2$, find$E\left[X\right].$

Two fair dice are rolled. Let $X$ equal the product of the $2$ dice. Compute $P\{X=i\}\text{for}i=1,鈥�,36$.