Expectation. Variance. Moments

Show that the integral $\int_{-\infty }^{+\infty } \left|x - m\right| f(x)\, dx$ becomes minimum when $m$ is the median of the distribution with density $f$.

Let $f(x)$ denote the density function of the random variable $X$. Suppose that $X$ has a symmetric distribution about $a$, that is, $f(x + a) = f(a - x)$ for every $x$. Show that the mean $\mathbb {E}\left[X\right]$ equals $a$, provided it exists.

Show that $\mathbb {E}\left[XY\right]^{2} \leq \mathbb {E}\left[X^{2}\right] \mathbb {E}\left[Y^{2}\right]$ provided the second moments exist.

Show that $\left|\mathbb {E}\left[X\right]\right| \leq \mathbb {E}\left[\left|X\right|\right]$.

For every set $A$ of real numbers, we define the indicator function by

Show that

If $\mathbb {E}\left[X\right] = \mathbb {E}\left[X^{2}\right] = 0$, show that $\mathbb {P}\left(X = 0\right) = 1$.

Show that the mean $\mu$ of a random variable $X$ has the property

Show that for a continuous random variable $X$ with density function $f$ and cumulative distribution function $F$

Consequently, in the graph below $\mu = \operatorname {area}(A) - \operatorname {area}(B)$.

If the $n$th moment of the random variable $X$ with distribution function $F$ exists, show that

Show that the first $n$ moments determine the first $n$ central moments and, conversely, that the first $n$ central moments with the mean determine the first $n$ moments.

If $X$ is bounded, i.e., there is a constant $k < \infty$ such that $\mathbb {P}\left(\left|X\right| \leq k\right) = 1$, then $X$ has moments of every order.

If the $n$-order moment $\mu_{n}'$ exists, then show that there exist all $\mu_{k}'$ ($k = 1, 2, \ldots , n - 1$).

Show that a necessary condition for the mean of a random variable with distribution function $F$ to exist is that

In a lottery, $n$ numbers are selected from the $N$ numbers $1, 2, \ldots , N$. Find the variance of the sum $S_{n}$ of the selected numbers.

An urn contains $N_{1}$ white balls and $N_{2}$ black balls; $n$ balls are drawn at random,

A student takes a multiple-choice test consisting of two problems. The first one has 3 possible answers and the second one has 5. The student chooses, at random, one answer as the right one from each of the two problems. Find:

In a lottery that sells 3,000 tickets the first lot wins $1,000, the second $500, and five other lots that come next win $100 each. What is the expected gain of a man who pays 1 dollar to buy a ticket?

A die is thrown until the result ``ace or even number'' appears three times. Find the expected number of throws:

$A$ pays 1 dollar for each participation in the following game: three dice are thrown; if one ace appears he gets 1 dollar, if two aces appear he gets 2 dollars and if three aces appear he gets 8 dollars; otherwise he gets nothing. Is the game fair, i.e., is the expected gain of the player zero? If not, how much should the player receive when three aces appear to make the game fair?

A player has 15 dollars. If heads appear on the first toss he receives 1 dollar and he withdraws from the game. If tails appear he bets 2 dollars and if he wins the second toss he withdraws; otherwise he continues playing on a bet of 4 dollars. If he loses, he bets the remaining amount of 8 dollars. What is the expected gain of the player?

Of the parcels mailed abroad 10 never reach their destination. Two books may be sent separately or in a single parcel. Each book is worth $2. The postage for each book sent separately is 10¢ and for both books in a single parcel 15¢. For each of the two ways of mailing, find:

Let $X$ be a random variable with distribution function

Plot the graph of $F(x)$ and then evaluate $\mathbb {E}\left[X\right]$. Give an example of a random variable $X$ that has the above distribution.

If the random variable $X$ is $\mathscr {N}\left(\mu , 1\right)$, show that the random variable $Y = [1 - \Phi (X)]/\varphi (X)$, where $\Phi$ and $\varphi$ denote the distribution function and the density of $\mathscr {N}\left(0, 1\right)$, respectively, has mean value $1/\mu$.

The truncated Poisson distribution with the zero class missing has probability function

Find $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.