Random Variables and Their Distributions

Show that $X: \Omega \rightarrow \mathbb {R}$ is a random variable if and only if the inverse image $\{ X \in B\}$ belongs to $\mathcal{F}$ for each Borel set $B \subset \mathbb {R}$.

If $X$ denotes the amount won (or lost) in the game described at the beginning of this chapter, what are the probabilities $\mathbb {P}\left(\{ X>0\} \right)$ and $\mathbb {P}\left(\{ X<0\} \right)$?

Two dice are rolled. Let $X$ be the larger of the two numbers shown. Compute $\mathbb {P}\left(\{ X \in [2,4]\} \right)$.

Let $\Omega =[0,1]$ with Borel sets and Lebesgue measure. Find $\mathbb {P}\left(\{ X \in \left[0, \frac{1}{2}\right)\} \right)$ for $X(x)=x^{2}$.

Let $X$ be the number of tosses of a fair coin up to and including the first toss showing heads. Find $\mathbb {P}\left(\{ X \in 2 \mathbb {N}\} \right)$, where $2 \mathbb {N}=\{ 2 n: n= 1,2, \ldots \}$ is the set of even integers.

Show that if $X$ is a constant function, then it is a random variable with respect to any $\sigma$-field.

Let $\Omega =\{ 1,2,3,4\}$ and $\mathcal{F}=\{ \varnothing , \Omega ,\{ 1\} ,\{ 2,3,4\} \}$. Is $X(\omega )=1+\omega$ a random variable with respect to the $\sigma$-field $\mathcal{F}$? If not, give an example of a non-constant function which is.

For each of the functions below, find the smallest $\sigma$-field on $\Omega = \{ -2,-1,0,1,2\}$ with respect to which the function is a random variable:

Find the smallest $\sigma$-fields on $\Omega =\{ -2,-1,0,1,2\}$ such that the following functions are random variables:

Consider $\Omega =[0,1]$ with the $\sigma$-field $\mathcal{F}$ of Borel sets $B$ contained in $[0,1]$.

Let $\Omega =[0,1]$ with the $\sigma$-field $\mathcal{G}$ of Borel sets $B$ contained in $[0,1]$ such that $B=1-B$. (By $1-B$ we denote the set $\{ 1-x: x \in B\}$.)

What is the smallest number of elements of a $\sigma$-field if a function $X: \Omega \rightarrow \mathbb {R}$ taking exactly $n$ different values is to be a random variable with respect to this $\sigma$-field?

Show that if $X$ is a random variable with respect to a $\sigma$-field $\mathcal{F}$ and $\mathcal{F} \subset \mathcal{G}$ for some $\sigma$-field $\mathcal{G}$, then $X$ is a random variable with respect to $\mathcal{G}$.

Let $X$ be a random variable. Show that the map

is a probability measure on the $\sigma$-field $\mathcal{B}$ of Borel sets in $\mathbb {R}$.

Find an example of two different random variables $X$ and $X^{\prime }$ with the same distribution $P_{X}=P_{X^{\prime }}$.

Let $X$ be the amount won (or lost) in the game described at the beginning of this chapter. Find $\mathbb {P}_{X}\left((0,+\infty )\right)$ and $\mathbb {P}_{X}\left((-\infty , 0)\right)$.

Two dice are rolled. Let $X$ be the larger of the two numbers shown. Compute $\mathbb {P}_{X}\left([2,4]\right)$.

Let $\Omega =[0,1]$ with Borel sets and Lebesgue measure. Find $\mathbb {P}_{X}\left(\left[0, \frac{1}{2}\right)\right)$ if $X(x)=x^{2}$.

Let $X$ be the number of tosses of a fair coin up to and including the first toss showing heads. Find $\mathbb {P}_{X}\left(2 \mathbb {N}\right)$, where $2 \mathbb {N}=\{ 2 n: n= 1,2, \ldots \}$ is the set of even integers.

$X(\omega ) \equiv c$ (a constant random variable identically equal to $c$).

$X(\omega )=1$ for $\omega \in A$ and $X(\omega )=2$ otherwise, where $\mathbb {P}\left(A\right)=\frac{1}{3}$.

$X(\omega )=c_{k}$ with probability $\alpha_{k}$ for $k=1, \ldots , n$, where $c_{1}<c_{2}< \cdots <c_{n}$ and $\alpha_{1}+\cdots +\alpha_{n}=1$.

$X(\omega )=1$ for $\omega \in A, X(\omega )=-2$ for $\omega \in B$, and $X(\omega )=2$ otherwise, where $\mathbb {P}\left(A\right)=\frac{1}{3}, \mathbb {P}\left(B\right)=\frac{1}{2}$, and $A \cap B=\varnothing$.

$\Omega =[0,1]$ with Lebesgue measure, $X(x)=2 x-1$.

Let $X$ be the number of tosses of a fair coin up to and including the first toss showing heads. Compute and sketch the distribution function $F_{X}$.

$F_{X}$ is non-decreasing, i.e., if $y_{1} \leqslant y_{2}$, then $F_{X}\left(y_{1}\right) \leqslant F_{X}\left(y_{2}\right)$.

$\lim_{y \rightarrow +\infty } F_{X}(y)=1$.

$\lim_{y \rightarrow -\infty } F_{X}(y)=0$.

Show that if all the values of $X$ are in an interval $[a, b]$, where $a<b$, then $F_{X}(y)=0$ for all $y<a$ and $F_{X}(y)=1$ for all $y \geqslant b$.

$F_{X}$ is right-continuous.

$F_{X}$ is continuous at $y$ if and only if $P_{X}(\{ y\} )=0$.

Let $a<b$. Find expressions for the following probabilities in terms of $F_{X}$:

Find $F_{X}$ if $\Omega =[0,1] \times [0,1]$ is the unit square with uniform measure (which measures the surface area) and $X(\omega )$ is the distance between $\omega \in \Omega$ and the nearest edge of the square.

A coin is tossed. If it shows heads, you pay 2 dollars. If it shows tails, you spin a wheel which gives the amount you win distributed with uniform probability between 0 and 10 dollars. Your gain (or loss) is a random variable $X$. Find the distribution function and use it to compute the probability that you will not win at least 5 dollars.

Your car has broken down at noon. You have called a repair team, which left at 12:00 from a place 100 miles from where you are. They travel at 100 miles per hour. In the meantime, you try to fix the car yourself. The chances of success are uniform in the time interval between 12 and 1 o'clock with total probability $\frac{1}{2}$. If you succeed, you will drive to meet the repair team at 100 miles per hour. Let $X$ be the distance traveled by the team before they meet you (the fee you pay depends on it). Find $F_{X}$.

Two towns A and B are 50 miles apart. Car 1 leaves town A and car 2 leaves town B independently of one another. The departure time of each car is random and uniformly distributed between 12 and 1 o'clock. The speed of each car is 100 miles per hour and they travel directly to meet one another. Denote by $X$ the distance between the meeting point and A. Find $F_{X}$.

Show that $X$ is a discrete random variable if and only if the distribution $P_{X}$ is a discrete probability measure on $\mathbb {R}$ in the sense of Definition 6.7.

A random variable $X$ is said to have the binomial distribution $B(n, p)$, where $n=1,2, \ldots$ and $p \in [0,1]$, if

for $k=0,1, \ldots , n$. Verify that such a random variable is discrete.

Show that the indicator function $\; \mathbb {1}_{A}$ of an event $A$ has the binomial distribution $B(n, p)$ with $n=1$ and $p=\mathbb {P}\left(A\right)$. (This is sometimes called the Bernoulli distribution.)

Let $X$ have the binomial distribution $B(n, p)$. Show that for any $k=1, \ldots , n$,

(This formula can be used in practice to compute $\mathbb {P}_{X}\left(\{ k\} \right)$ by induction starting from $\mathbb {P}_{X}\left(\{ 0\} \right)=(1-p)^{n}$ or by backward induction starting from $\mathbb {P}_{X}\left(\{ n\} \right)=p^{n}$.)

Let $X$ have the binomial distribution $B(n, p)$ and let $Y$ have the binomial distribution $B(n, 1-p)$. Verify that for any $k=0,1, \ldots , n$,

A biased coin with probability of heads $p$ and tails $1-p$ is flipped $n$ times. Let $X$ denote the number of heads obtained. What is the distribution of $X$?

A biased coin with probability of heads $p$ and tails $1-p$ is flipped until $k$ heads are obtained.

Let $X_{n}$ be a sequence of random variables with binomial distribution $B\left(n, \frac{\lambda }{n}\right)$ for some $\lambda >0$. Show that

A random variable $X$ is said to have the Poisson distribution with parameter $\lambda >0$ if

for $k=0,1,2, \ldots$. Show that $X$ is a discrete random variable.

A safety device in a laboratory is set to activate an alarm if it registers five or more radioactive particles within one second. If the background radiation is such that the number of particles reaching the device has the Poisson distribution with parameter $\lambda =0.5$, how likely is it that the alarm will be activated within a given 1-second period?

Let $X$ have the Poisson distribution with parameter $\lambda$. Verify that

for each $k=0,1,2, \ldots$. (This formula can be used to compute $\mathbb {P}_{X}\left(\{ k\} \right)$ by induction starting from $\mathbb {P}_{X}\left(\{ 0\} \right)=e^{-\lambda }$.)

Show that $\int_{-\infty }^{\infty } f_{X}(x) \mathrm{d} x=1$.

Show that $P_{X}(\{ y\} )=0$ for any $y \in \mathbb {R}$.

Give a formula for $F_{X}(y)$ in terms of $f_{X}$.

Show that for any absolutely continuous random variable, the distribution function is continuous.

Show that if the distribution function $F_{X}$ is differentiable with continuous derivative, then $X$ is an absolutely continuous random variable with density $f_{X}(x)=\frac{d F_{X}(x)}{d x}$.

A random variable $X$ has the normal distribution $N\left(m, \sigma^{2}\right)$, where $m, \sigma \in \mathbb {R}$, if it is an absolutely continuous random variable with density

Verify that $f_{X}$ is indeed a density.

Let $X$ be a random variable with normal distribution $N(0,1)$. Show that $a+b X$, where $a, b \in \mathbb {R}$, has the normal distribution $N\left(a, b^{2}\right)$.

Let $X$ be a random variable with normal distribution $N(0,1)$. Find the density of $Y=e^{X}$. (The distribution of $Y$ is called the log normal distribution.)

Let $X$ be an absolutely continuous random variable with uniform distribution on $[0,1]$, i.e., with $f_{X}(x)=1$ for any $x \in [0,1]$ and $f_{X}(x)=0$ otherwise. Find a function $g: \mathbb {R} \rightarrow \mathbb {R}$ such that $g(X)$ has the normal distribution $N(0,1)$.

Suppose that $X$ and $Y$ are random variables defined on the same probability space $(\Omega , \mathcal{F}, P)$. Show that $\{ (X, Y) \in B\}$ belongs to $\mathcal{F}$ for any Borel set $B$ in $\mathbb {R}^{2}$.

Let $X$ and $Y$ be random variables defined on the same probability space. Show that the distribution $P_{X}$ of $X$ and $P_{Y}$ of $Y$ can be obtained from the joint distribution as follows:

for any Borel set $B$ in $\mathbb {R}$.

Two identical coins are flipped simultaneously. Let $X$ be the number of heads and $Y$ the number of tails shown. What is the joint distribution of $X$ and $Y$? What are the marginal distributions?

A pair of dice are rolled. Let $X$ be the greater and $Y$ the smaller of the numbers shown. Find the joint distribution and the marginal distributions of $X$ and $Y$.

Two balls are drawn from an urn containing one yellow, two red and three blue balls. Let $X$ be the number of red balls and $Y$ the number of blue balls drawn. Find the joint distribution and the marginal distributions of $X$ and $Y$.

Prove that random variables $X$ and $Y$ are jointly discrete if and only if they are both discrete.

A fair coin is tossed repeatedly. Suppose that heads appear for the first time after $X$ tosses and tails appear for the first time after $Y$ tosses. Show that $X$ and $Y$ are jointly discrete random variables and find their joint distribution. Hence or otherwise, compute the marginal distributions of $X$ and $Y$.

Let $X$ and $Y$ have the joint distribution

for $m, n=1,2, \ldots$. Compute the marginal distributions $P_{X}$ and $P_{Y}$.

Show that if $X$ and $Y$ are jointly continuous random variables, then both $X$ and $Y$ are absolutely continuous with density, respectively,

If expressed in this way, $f_{X}$ and $f_{Y}$ are called the marginal densities.

Find two absolutely continuous random variables $X$ and $Y$ that are not jointly continuous.

If $X$ and $Y$ are jointly continuous random variables, show that $U= X+Y$ and $V=X-Y$ are also jointly continuous with joint density

Show that $X$ and $Y$ are independent random variables if and only if the events $\{ X \in A\}$ and $\{ Y \in B\}$ are independent for any Borel sets $A, B \subset \mathbb {R}$.

Let $X$ and $Y$ be independent random variables. Show that $f(X)$ and $g(Y)$ are independent for any Borel functions $f$ and $g$. (By $f(X)$, we denote the random variable $\Omega \ni \omega \mapsto f(X(\omega )) \in \mathbb {R}$ and by $g(Y)$ the random variable $\Omega \ni \omega \mapsto g(Y(\omega )) \in \mathbb {R}$.)

Show that jointly continuous random variables are independent if and only if for every $x, y \in \mathbb {R}$,

Suppose that $X$ and $Y$ are jointly continuous random variables with joint density $f_{X, Y}(x, y)=c e^{x+y}$ for $x, y \in (-\infty , 0]$ and $f_{X, Y}(x, y)= 0$ otherwise.

Let $X$ and $Y$ be independent random variables with binomial distribution $B(m, p)$ and $B(n, p)$, respectively. What is the distribution of $X+Y$?

Let $X$ and $Y$ be independent random variables having the Poisson distribution with parameter $\lambda$ and $\mu$, respectively. Find the distribution of $X+Y$.

Suppose that $X$ and $Y$ are jointly continuous independent random variables. Show that the density of the sum $X+Y$ is the convolution of the marginal densities: