Continuous random variables

For what values of the parameters are the following functions probability density functions?

Find the density function of $Y = a X$, where $a > 0$, in terms of the density function of $X$. Show that the continuous random variables $X$ and $-X$ have the same distribution function if and only if $f_{X}(x) = f_{X}(-x)$ for all $x \in \mathbb {R}$.

If $f$ and $g$ are density functions of random variables $X$ and $Y$, show that $\alpha f+(1-\alpha ) g$ is a density function for $0 \leq \alpha \leq 1$, and describe a random variable of which it is the density function.

Let $X$ be a positive random variable with density function $f$ and distribution function $F$. Define the hazard function $H(x) = -\log [1-F(x)]$ and the hazard rate

Show that:

I am selling my house, and have decided to accept the first offer exceeding $£ K$. Assuming that offers are independent random variables with common distribution function $F$, find the expected number of offers received before I sell the house.

Let $X$ and $Y$ be independent random variables with common distribution function $F$ and density function $f$. Show that $V = \max \left\{ X, Y\right\}$ has distribution function $\mathbb {P}\left(V \leq x\right) = F(x)^{2}$ and density function $f_{V}(x) = 2 f(x) F(x), x \in \mathbb {R}$. Find the density function of $U = \min \left\{ X, Y\right\}$.

The annual rainfall figures in Bandrika are independent identically distributed continuous random variables $\left\{ X_{r}: r \geq 1\right\}$. Find the probability that:

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) < 1$ for all $y$, and let $Y(y) = \min \left\{ k: X_{k} > y\right\}$. Show that

Let $P_{i} = \left(X_{i}, Y_{i}\right), 1 \leq i \leq n$, be independent, uniformly distributed points in the unit square $[0,1]^{2}$. A point $P_{i}$ is called peripheral if, for all $r = 1,2, \ldots , n$, either $X_{r} \leq X_{i}$ or $Y_{r} \leq Y_{i}$, or both. Show that the mean number of peripheral points is $n\left(\frac{3}{4}\right)^{n-1}$.

Let $U$ and $V$ be independent and uniformly distributed on the interval $[0,1]$.

For what values of $\alpha$ is $\mathbb {E}\left[|X|^{\alpha }\right]$ finite, if the density function of $X$ is:

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent identically distributed random variables, and let $S_{m} = X_{1}+ X_{2}+\cdots +X_{m}$. Assume that $\mathbb {P}\left(S_{n} = 0\right) = 0$. Show that, if $m \leq n$, then $\mathbb {E}\left[S_{m} / S_{n}\right] = m / n$.

Let $X$ be a non-negative random variable with density function $f$. Show that

for any $r \geq 1$ for which the expectation is finite.

Show that the mean $\mu$, median $m$, and variance $\sigma^{2}$ of the continuous random variable $X$ satisfy $(\mu -m)^{2} \leq \sigma^{2}$.

Let $X$ be a random variable with mean $\mu$ and continuous distribution function $F$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be non-negative random variables with finite means. Show that

If $X$ is a continuous random variable and $\mathbb {E}\left[X^{r}\right]$ exists, where $r \geq 1$ is an integer, show that

Let $X$ be a random variable taking values in $[0, a]$, with density function $f$ satisfying $f(x) \leq b$ for all $x$. Show that $\mathbb {E}\left[X\right] \geq (2 b)^{-1}$. More generally, show for $n \geq 1$ that $\mathbb {E}\left[X^{n}\right] \geq 1 /\left[(n+1) b^{n}\right]$.

Let $X$ be continuous with finite variance. Show that $g(a) = \mathbb {E}\left[(X-a)^{2}\right]$ is a minimum when $a = \mathbb {E}\left[X\right]$.

A target comprising a disc with centre $O$ and unit radius is hit by $n$ arrows. They strike independent spots on the target, and for each the chance of hitting within distance $a \in (0,1)$ of O is $a^{2}$. Let $R$ be the radius of the smallest circle centred at O that covers all the arrow strikes. Show that $R$ has density $f(r) = 2 n r^{2 n-1}, 0 \leq r \leq 1$, and find the mean area of this circle.

The arrow furthest from $O$ falls out. Show that the area of the smallest circle centred at $O$ and covering the remaining arrows is $(n-1) \pi /(n+1)$.

Let $X$ be continuous with variance $\sigma^{2}$, and also unimodal about a mode $M$. Show that $\left|M-\mathbb {E}\left[X\right]\right| \leq \sigma \sqrt{3}$, with equality if and only if $X$ is uniformly distributed.

What can you say about $\left|M-m\right|$ where $m$ is a median of $X$?

A football is placed at a uniformly distributed position $W$ in the unit interval $[0,1]$, and kicked with a non-zero random velocity $V$ having a distribution that is symmetric about $0$. Assuming that $V$ and $W$ are independent, show (neglecting air resistance) that the time $T$ at which the football hits an endpoint of the interval is unimodal.

Here are two further questions concerning the football of the previous exercise.

Prove that the gamma function satisfies $\Gamma (t) = (t-1) \Gamma (t-1)$ for $t > 1$, and deduce that $\Gamma (n) = (n-1)!$ for $n = 1,2, \ldots$ Show that $\Gamma \left(\frac{1}{2}\right) = \sqrt{\pi }$ and deduce a closed form for $\Gamma \left(n+\frac{1}{2}\right)$ for $n = 0,1,2, \ldots$.

Show, as in paragraph (4.4.8), that the beta function satisfies $B(a, b) = \Gamma (a) \Gamma (b) / \Gamma (a+b)$.

Let $X$ have the uniform distribution on $[0,1]$. For what function $g$ does $Y = g(X)$ have the exponential distribution with parameter 1?

Find the distribution function of a random variable $X$ with the Cauchy distribution. For what values of $\alpha$ does $\left|X\right|$ have a finite (possibly fractional) moment of order $\alpha$?

Let $Y = e^{X}$ where $X$ has the $N(0,1)$ distribution. Find the density function of $Y$.

Let $X$ be $N\left(\mu , \sigma^{2}\right)$. Show that $\mathbb {E}\left[(X-\mu ) g(X)\right] = \sigma^{2} \mathbb {E}\left[g^{\prime }(X)\right]$ when both sides exist.

With the terminology of Exercise (4.1.4), find the hazard rate when:

Let $X_1, X_2, X_3$ be independent, exponentially distributed random variables with respective parameters $\lambda_1, \lambda_2, \lambda_3$. Show that

Let $U$ and $X$ be independent, where $U$ is uniform on $(0,1)$ and $X$ is exponentially distributed with parameter $\lambda$. Show that $\mathbb {E}\left[\min \left\{ U, X\right\} \right] = \lambda^{-1} e^{-\lambda }-\lambda^{-2}\left(1-e^{-\lambda }\right)$.

Let $X$ have the Cauchy distribution, and show that $Y = 1 /\left(1+X^{2}\right)$ has the arc sine density function

Let $X_{1}, X_{2}, \ldots$ be independent and uniformly distributed on $[-c, c]$.

Let $X$ have the beta distribution $\beta (a, b)$. Show that $Y = X /(1-X)$ has density function

This is called the beta distribution of the second kind. Find $\mathbb {E}\left[Y^{n}\right]$ where $1 \leq n < b$.

Let $Z$ have the $\Gamma (1, t)$ distribution of paragraph (4.4.6). Show that

Let

Show that $f$ is a continuous joint density function, but that the (first) marginal density function $g(x) = \int_{-\infty }^{\infty } f(x, y) d y$ is not continuous.

Let $Q = \left\{ q_{n}: n \geq 1\right\}$ be a set of real numbers, and define

Show that $f_{Q}$ is a continuous joint density function whose first marginal density function is discontinuous at the points in $Q$. Can you construct a continuous joint density function whose first marginal density function is continuous nowhere?

Two grids of parallel lines are superimposed: the first grid contains lines distance $a$ apart, and the second contains lines distance $b$ apart which are perpendicular to those of the first set. A needle of length $r( < \min \left\{ a, b\right\} )$ is dropped at random. Show that the probability it intersects a line equals $r(2 a+2 b-r) /(\pi a b)$.

The plane is ruled by the lines $y = n$, for $n = 0, \pm 1, \ldots$, and onto this plane we drop a cross formed by welding together two unit needles perpendicularly at their midpoints.

Let $X$ and $Y$ be independent random variables each having the uniform distribution on $[0,1]$. Let $U = \min \left\{ X, Y\right\}$ and $V = \max \left\{ X, Y\right\}$. Find $\mathbb {E}\left[U\right]$, and hence calculate $\operatorname {Cov}\left[U, V\right]$.

Three points A, B, C are chosen independently at random on the circumference of a circle. Let $b(x)$ be the probability that at least one of the angles of the triangle ABC exceeds $x \pi$. Show that

Hence find the density and expectation of the largest angle in the triangle.

Let $\left\{ X_{r}: 1 \leq r \leq n\right\}$ be independent and identically distributed with finite variance, and define $\bar{X} = n^{-1} \sum_{r = 1}^{n} X_{r}$. Show that $\operatorname {Cov}\left[\bar{X}, X_{r}-\bar{X}\right] = 0$.

Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X+Y$ and $V = X Y$. Under what condition are $U$ and $V$ uncorrelated?

Let $X$ and $Y$ be independent continuous random variables, and let $U$ be independent of $X$ and $Y$ taking the values $\pm 1$ with probability $\frac{1}{2}$. Define $S = U X$ and $T = U Y$. Show that $S$ and $T$ are in general dependent, but $S^{2}$ and $T^{2}$ are independent.

Let $X, Y, Z$ be independent and identically distributed continuous random variables. Show that $\mathbb {P}\left(X > Y\right) = \mathbb {P}\left(Z > Y\right) = \frac{1}{2}$. What is $\mathbb {P}\left(Z > Y \mid X > Y\right)$?

Let $(X, Y)$ and $(U, V)$ be independent random vectors with common distribution function $F(x, y)$ and marginal distribution functions $F_{X}(x)$ and $F_{Y}(y)$. Show that, when $\left|\operatorname {Cov}\left[X, Y\right]\right| < \infty$, we have that $\mathbb {E}\left[(X-U)(Y-V)\right] = 2 \operatorname {Cov}\left[X, Y\right]$ and

Let the pair $X, Y$ have the bivariate normal distribution with means $0$, variances $1$, and correlation $\rho$. Show that

Let $X$ be exponentially distributed with parameter $\lambda$. Let $N$ be the greatest integer not greater than $X$, and set $M = X-N$. Show that $M$ and $N$ are independent. Find the density function of $M$ and the distribution of $N$.

Let $X$ and $Y$ be independent $N(0,1)$ random variables. Show, if $2 a < 1,2 b < 1$, and $4 b^{2} < (1-2 a)(1-2 c)$, that

Let $X, Y$ be random variables with joint density function $f(x, y)$ and marginal densities $g, h$. Their contingency coefficient is given as

If $X, Y$ are bivariate normal with correlation $\rho \in (-1,1)$, show that $1+\phi^{2} = 1 /\left(1-\rho^{2}\right)$. Show in addition that their mutual information (see Exercise (3.6.5)) is $I = -\frac{1}{2} \log \left(1-\rho^{2}\right)$.

Let $X$ be uniformly distributed on $[-1,1]$. Are the random variables $Z_{n} = \cos (n \pi X), n = 1,2, \ldots$, correlated? Are they independent? Explain your answers.

The mean absolute difference, or MAD, of two independent random variables $X, Y$, with common distribution function $F$, is given by $\operatorname {MAD} = \mathbb {E}\left[\left|X-Y\right|\right]$. Show that, for non-negative random variables $X, Y$,

More generally, for $X$ and $Y$ taking values in $\mathbb {R}$, with a common continuous distribution function with unique inverse $Q$, show that

Find $\operatorname {MAD}$ for the following distributions:

Let $X, Y$ be independent $N(0,1)$ random variables, and $U = \min \left\{ \left|X\right|,\left|Y\right|\right\}$, $V = \max \left\{ \left|X\right|,\left|Y\right|\right\}$. Show that $\mathbb {E}\left[U / V\right] = (4 / \pi ) \log \sqrt{2}$.

A point $P = (X, Y)$ is selected uniformly at random inside a triangle $\Delta$ with corners $(1,0),(1,1),(0,1)$. Show that a triangle ABC with $\mathrm{BC} = X, \mathrm{CA} = Y, \mathrm{AB} = 2-X-Y$ can always be constructed, and prove that the angle $\widehat{\mathrm{ABC}}$ is obtuse with probability $3-4 \log 2$.

Random variables $X, Y$ have joint density function $f(x, y) = 3\left((x+y)-\left(x^{2}+y^{2}\right)\right)$ for $x, y \in [0,1]$. Show that $X$ and $Y$ are uncorrelated but dependent.

A point is picked uniformly at random on the surface of a unit sphere. Writing $\Theta$ and $\Phi$ for its longitude and latitude, find the conditional density functions of $\Theta$ given $\Phi$, and of $\Phi$ given $\Theta$.

Show that the conditional expectation $\psi (X) = \mathbb {E}\left[Y \mid X\right]$ satisfies $\mathbb {E}\left[\psi (X) g(X)\right] = \mathbb {E}\left[Y g(X)\right]$, for any function $g$ for which both expectations exist.

Construct an example of two random variables $X$ and $Y$ for which $\mathbb {E}\left[Y\right] = \infty$ but such that $\mathbb {E}\left[Y \mid X\right] < \infty$ almost surely.

Find the conditional density function and expectation of $Y$ given $X$ when they have joint density function:

Let $Y$ be distributed as $\operatorname {bin}(n, X)$, where $X$ is a random variable having a beta distribution on $[0,1]$ with parameters $a$ and $b$. Describe the distribution of $Y$, and find its mean and variance. What is the distribution of $Y$ in the special case when $X$ is uniform?

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent and uniformly distributed on $[0,1]$. Let $0 < x < 1$ and define

Show that $\mathbb {P}\left(N > n\right) = x^{n} / n!$, and hence find the mean and variance of $N$.

Let $X$ and $Y$ be random variables with correlation $\rho$. Show that $\mathbb {E}\left[\operatorname {Var}\left[Y \mid X\right]\right] \leq \left(1-\rho^{2}\right) \operatorname {Var}\left[Y\right]$.

Let $X, Y, Z$ be independent and exponential random variables with respective parameters $\lambda , \mu , \nu$. Find $\mathbb {P}\left(X < Y < Z\right)$.

Let $X$ and $Y$ have the joint density $f(x, y) = c x(y-x) e^{-y}, 0 \leq x \leq y < \infty$.

Let $\left\{ X_{r}: r \geq 0\right\}$ be independent and identically distributed random variables with density function $f$ and distribution function $F$. Let $N = \min \left\{ n \geq 1: X_{n} > X_{0}\right\}$ and $M = \min \left\{ n \geq 1 : X_{0} \geq X_{1} \geq \cdots \geq X_{n-1} < X_{n}\right\}$. Show that $X_{N}$ has distribution function $F+(1-F) \log (1-F)$, and find $\mathbb {P}\left(M = m\right)$.

Let the point $P = (X, Y)$ be uniformly distributed in the square $S = [0,1]^{2}$, and denote by $f(x \mid D)$ the density function of $X$ conditional on the event $D$ that $P$ lies on the diagonal of $S$ through the origin.