Multivariate Random Variables

Show that if $X \in C(0,1)$, then so is $1/X$.

Let $X \in C(m,a)$. Determine the distribution of $1/X$.

Show that if $T \in t(n)$, then $T^{2} \in F(1,n)$.

Show that if $F \in F(m,n)$, then $1/F \in F(n,m)$.

Show that if $X \in C(0,1)$, then $X^{2} \in F(1,1)$.

Show that $\beta (1,1) = U(0,1)$.

Show that if $F \in F(m,n)$, then $1/\left(1+\frac{m}{n}F\right) \in \beta (n/2,m/2)$.

Show that if $X$ and $Y$ are independent $N(0,1)$-distributed random variables, then $X/Y \in C(0,1)$.

Show that if $X \in N(0,1)$ and $Y \in \chi^{2}(n)$ are independent random variables, then $X/\sqrt{Y/n} \in t(n)$.

Show that if $X \in \chi^{2}(m)$ and $Y \in \chi^{2}(n)$ are independent random variables, then $(X/m)/(Y/n) \in F(m,n)$.

Show that if $X$ and $Y$ are independent $\mathrm{Exp}(a)$-distributed random variables, then $X/Y \in F(2,2)$.

Let $X$ and $Y$ be independent random variables such that $X \in U(0,1)$ and $Y \in U(0,\alpha )$. Find the density function of $Z = X+Y$.

Let $X$ and $Y$ have a joint density function given by

Determine the marginal density functions and the joint and marginal distribution functions.

Suppose that $X \in \mathrm{Exp}(1)$, let $Y$ be the integer part and $Z$ the fractional part, that is, let

Show that $Y$ and $Z$ are independent and find their distributions.

Ottar jogs regularly. One day he started his run at 5:31 p.m.and returned at 5:46 p.m. The following day he started at 5:31 p.m.and returned at 5:47 p.m. His watch shows only hours and minutes (not seconds). What is the probability that the run the first day lasted longer than the run the second day?

A certain chemistry problem involves the numerical study of a lognormal random variable $X$. Suppose that the software package used requires the input of $\mathbb {E}\left[Y\right]$ and $\operatorname {Var}\left[Y\right]$ into the computer (where $Y$ is normal and such that $X = e^{Y}$), but that one knows only the values of $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$. Find expressions for the former mean and variance in terms of the latter.

Let $X$ and $Y$ be independent $\mathrm{Exp}(a)$-distributed random variables. Find the density function of the random variable $Z = X/(1+Y)$.

Let $X \in \mathrm{Exp}(1)$ and $Y \in U(0,1)$ be independent random variables. Determine the distribution (density) of $X+Y$.

The random vector $\overrightarrow {X} = (X_{1}, X_{2}, X_{3})'$ has density function

Determine the distribution of $X_{1} \cdot X_{2} \cdot X_{3}$.

The random variables $X_{1}$ and $X_{2}$ are independent and equidistributed with density function

Set $Y_{1} = X_{1}\sqrt{X_{2}}$ and $Y_{2} = X_{2}\sqrt{X_{1}}$.

Let $(X,Y)'$ have density

Show that $X$ and $X \cdot Y$ are independent, equidistributed random variables and determine their distribution.

Let $X$ and $Y$ have joint density

Determine the distribution of $Y-X$.

Suppose that $(X,Y)'$ has a density function given by

Determine the distribution of $X^{2}Y$.

Let $X$ and $Y$ have the following joint density function:

Show that $Y$ and $X/(Y-X)$ are independent, and find their distributions.

Let $X$ and $Y$ have joint density

Determine the distribution of $XY$.

Suppose that $X$ and $Y$ are random variables with a joint density

Show that $X/Y$ and $Y$ are independent standard exponential random variables and exploit this fact in order to compute $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.

Let $X$ and $Y$ have joint density

Determine the distribution of $XY$.

Let $X$ and $Y$ have joint density

Determine the distribution of $X^{2}/Y$.

Suppose that $(X,Y)'$ has density

Determine the distribution of

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution of $2X+3Y$.

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution of $X(1+Y)$.

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution of $X/(1+Y)^{2}$.

Suppose that $X$, $Y$, and $Z$ are random variables with a joint density

Determine the distribution of $X+Y+Z$.

Suppose that $X$, $Y$, and $Z$ are random variables with a joint density

Determine the distribution of $X+Y$.

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution of $X-Y$.

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution of $Y/X$.

Suppose that $X$ and $Y$ are independent $\mathrm{Pa}(1,1)$-distributed random variables. Determine the distributions of $XY$ and $X/Y$.

Suppose that $X$ and $Y$ are random variables with a joint density

Determine the distribution (density) of $Z = -\log (Y/X)$.

Let $X_{1} \in \Gamma (a_{1},b)$ and $X_{2} \in \Gamma (a_{2},b)$ be independent random variables. Show that $X_{1}/X_{2}$ and $X_{1}+X_{2}$ are independent random variables, and determine their distributions.

Let $X \in \Gamma (r,1)$ and $Y \in \Gamma (s,1)$ be independent random variables.

Let $X_{1}$, $X_{2}$, and $X_{3}$ be independent random variables, and suppose that $X_{i} \in \Gamma (r_{i},1)$, $i=1,2,3$. Set

Determine the joint distribution of $Y_{1}$, $Y_{2}$, and $Y_{3}$. Conclusions?

Let $X$ and $Y$ be independent $N(0,1)$-distributed random variables.

Let $X$ and $Y$ be independent random variables. Determine the distribution of $(X-Y)/(X+Y)$ if

A random vector in $\mathbb {R}^{2}$ is chosen as follows: Its length, $Z$, and its angle, $\Theta$, with the positive $x$-axis, are independent random variables, $Z$ has density

and $\Theta \in U(0,2\pi )$. Let $Q$ denote the point of the vector. Determine the joint distribution of the Cartesian coordinates of $Q$.

Show that the following procedure generates $N(0,1)$-distributed random numbers: Pick two independent $U(0,1)$-distributed numbers $U_{1}$ and $U_{2}$ and set $X = \sqrt{-2\log U_{1}} \cdot \cos (2\pi U_{2})$ and $Y = \sqrt{-2\log U_{1}} \cdot \sin (2\pi U_{2})$. Show that $X$ and $Y$ are independent $N(0,1)$-distributed random variables.