Expectation and Variance

Let $X$ be the number shown when a single die is rolled (the amount won in the game above). Find $\mathbb {E}\left[X\right]$.

What is the expectation of a random variable with Bernoulli distribution?

A friend of yours forgot the 8-digit password necessary to log into his computer. If he tries all possible passwords completely at random, discarding the unsuccessful ones, what is the expected number of attempts needed to find the correct password?

A die is rolled twice. What is the expectation of the sum of outcomes?

What is the expectation of a random variable with binomial distribution?

Linearity. Show that $\mathbb {E}\left[X+Y\right]=\mathbb {E}\left[X\right]+\mathbb {E}\left[Y\right]$ and $\mathbb {E}\left[c X\right]= c \mathbb {E}\left[X\right]$ for any simple random variables $X$ and $Y$ and any $c \in \mathbb {R}$.

Positivity. Show that if $X$ is a simple random variable and $X \geqslant 0$, then $\mathbb {E}\left[X\right] \geqslant 0$.

Monotonicity. If $X$ and $Y$ are simple random variables such that $X \leqslant Y$, then $\mathbb {E}\left[X\right] \leqslant \mathbb {E}\left[Y\right]$.

Show that $\left|\mathbb {E}\left[X\right]\right| \leqslant \mathbb {E}\left[\left|X\right|\right]$ for any simple random variable.

Let $X$ be a simple random variable such that $X \geqslant 0$. Show that if $\mathbb {E}\left[X\right]=0$, then $\mathbb {P}\left(\{ X=0\} \right)=1$.

Let $X$ be a simple random variable with $n$ different values $x_{1}, \ldots$, $x_{n}$. Show that for any function $g: \mathbb {R} \rightarrow \mathbb {R}$,

where $f(x)=\mathbb {P}\left(\{ X=x\} \right)$ is the mass function.

Let $X$ and $Y$ be simple random variables. Verify the Schwarz inequality

For simple random variables $X$ and $Y$, prove the triangle inequality

Prove the following Jensen inequality: If $X$ is a simple random variable with values in an interval $(a, b)$ and $h:(a, b) \rightarrow \mathbb {R}$ is a convex function, then

Show that $\mathbb {E}\left[\; \mathbb {1}_{A}\right]=\mathbb {P}\left(A\right)$ for any event $A$.

Verify the identity $1-\; \mathbb {1}_{A \cup B}=\left(1-\; \mathbb {1}_{A}\right)\left(1-\; \mathbb {1}_{B}\right)$ and use it to prove that $\mathbb {P}\left(A \cup B\right)=\mathbb {P}\left(A\right)+\mathbb {P}\left(B\right)-\mathbb {P}\left(A \cap B\right)$.

Verify the identity

and use it to prove the inclusion-exclusion formula stated in Solution 4.20.

For any independent simple random variables $X$ and $Y$, show that

Show that for any simple random variable $X$,

Compute the variance of a random variable with the Bernoulli distribution.

Show that if $\operatorname {Var}\left[X\right]=0$, then $\mathbb {P}\left(\{ X=a\} \right)=1$ for some $a \in \mathbb {R}$.

Show that $\operatorname {Var}\left[X+a\right]=\operatorname {Var}\left[X\right]$ for any $a \in \mathbb {R}$.

Show that $\operatorname {Var}\left[b X+a\right]=b^{2} \operatorname {Var}\left[X\right]$ for any $a, b \in \mathbb {R}$.

Show that

if $X$ and $Y$ are independent simple random variables.

Compute the variance of a random variable with the binomial distribution.

Find an example of a discrete random variable whose expectation is undefined.

A biased coin with probability of heads $p$ and tails $1-p$ is tossed repeatedly. Let $X$ be the number of tosses until heads appear for the first time. Compute the expectation of $X$. (The distribution of $X$ is called the geometric distribution.)

A coin is tossed repeatedly until a tail is obtained and you win $a^{k}$ dollars, where $k$ is the number of heads before the first tail and $a>0$. How much would you consider a fair amount to pay to play this game?

What is the expectation of a random variable with Poisson distribution?

Compute the expectation of a random variable with negative binomial distribution.

Let $X$ be a discrete random variable. Show that for any Borel function $g: \mathbb {R} \rightarrow \mathbb {R}$, the random variable $g(X)$ is also discrete and, provided that the series below is absolutely convergent,

where the $x_{1}, x_{2}, \ldots$ are the values of $X$ such that $x_{i} \neq x_{j}$ if $i \neq j$, and where $f(x)=\mathbb {P}\left(\{ X=x\} \right)$ is the mass function of $X$.

Let $X \geqslant 0$ be a discrete random variable and let $0 \leqslant X_{1} \leqslant X_{2} \leqslant \cdots$ be a non-decreasing sequence of discrete random variables such that $X_{n} \nearrow X$ as $n \rightarrow \infty$. Show that $\mathbb {E}\left[X_{n}\right] \nearrow \mathbb {E}\left[X\right]$ as $n \rightarrow \infty$.

Find an example of a discrete random variable whose expectation exists, but the variance is undefined.

Compute the variance of a random variable with the Poisson distribution.

Compute the variance of a random variable with the geometric distribution; see Problem 9.27.

Find an example of an absolutely continuous random variable $X$ whose expectation is undefined.

$X$ has the uniform distribution on $[a, b]$ with density $f_{X}(x)=\frac{1}{b-a}$ if $x \in [a, b]$ and 0 if $x \notin [a, b]$.

$X$ has the exponential distribution with density

$X$ has the Cauchy distribution with density

$X$ has the normal distribution with density

where $a \in \mathbb {R}$ and $b>0$ are given numbers.

Use Theorem 9.1 to compute the expectation of a random variable with log normal distribution (see Problem 8.55 for the definition of log normal distribution).

Use Theorem 9.1 to verify that

for any absolutely continuous random variable $X$, provided that both expectations on the right-hand side exist.

Compute the variance of a random variable with exponential distribution.

Compute the variance of a random variable with normal distribution.

$P_{X}(A)=\frac{1}{3} \delta_{0}(A)+\frac{2}{3} P_{2}(A)$, where $P_{2}$ is an absolutely continuous probability measure with density $f(x)=\frac{1}{2} \; \mathbb {1}_{[1,3]}$ (see Problem 6.39).

$P_{X}(A)=\frac{1}{8} \delta_{1}(A)+\frac{3}{8} \delta_{2}(A)+\frac{1}{2} P_{3}(A)$, where $P_{3}$ is an absolutely continuous probability measure with density $f(x)=\frac{1}{2} \; \mathbb {1}_{[1,3]}$ (see Problem 6.40).

$P_{X}(A)=\frac{1}{4} \delta_{0}(A)+\frac{1}{2} \delta_{2}(A)+\frac{1}{4} P_{3}(A)$, where $P_{3}$ is an absolutely continuous probability measure with density $f(x)=e^{-x} \; \mathbb {1}_{[0,+\infty )}$ (see Problem 6.41).

On your way to work you have to drive through a busy junction, where you may be stopped at traffic lights. The cycle of the traffic lights is 2 minutes of green followed by 3 minutes of red. What is the expected delay in the journey if you arrive at the junction at a random time uniformly distributed over the whole 5-minute cycle?

A fair coin is tossed repeatedly. For each occurrence of heads, you win $\frac{2}{3^{k}}$, where $k$ is the number of the toss. For tails, you win nothing. Show that the gain $X$ after infinitely many tosses is a singular random variable, i.e., neither discrete nor absolutely continuous nor a mixture thereof.

What is the expected gain $\mathbb {E}\left[X\right]$ after infinitely many tosses of the coin in Problem 9.49?