Math Problems Vault

Problem 4.8.1

Let $X$ and $Y$ be independent variables having the exponential distribution with parameters $\lambda$ and $\mu$ respectively. Find the density function of $X+Y$ .

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Problem 4.8.2

Let $X$ and $Y$ be independent variables with the Cauchy distribution. Find the density function of $\alpha X+\beta Y$ where $\alpha \beta \neq 0$ . (Do you know about contour integration?)

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Problem 4.8.3

Find the density function of $Z = X+Y$ when $X$ and $Y$ have joint density function $f(x, y) = \frac{1}{2}(x+y) e^{-(x+y)}, x, y \geq 0$ .

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Problem 4.8.4

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent exponential random variables with respective parameters $\left\{ \lambda_{r}: r \geq 1\right\}$ no two of which are equal. Find the density function of $S_{n} = \sum_{r = 1}^{n} X_{r}$ . [Hint: Use induction.]

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Problem 4.8.5

(a)(b)(c)

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(a)

Let $X, Y, Z$ be independent and uniformly distributed on $[0,1]$ . Find the density function of $X+Y+Z$ .

(b)

If $\left\{ X_{r}: r \geq 1\right\}$ are independent and uniformly distributed on $[0,1]$ , show that the density of $S_{n} = \sum_{r = 1}^{n} X_{r}$ at any point $x \in (0, n)$ is a polynomial in $x$ of degree $n-1$ . Show in particular that the density function $f_{n}$ of $S_{n}$ satisfies $f_{n}(x) = x^{n-1} /(n-1)$ ! for $x \in [0,1]$ .

(c)

Let $n \geq 3$ . What is the probability that the $X_{1}, X_{2}, \ldots , X_{n}$ of part (b) can be the lengths of the edges of an $n$ -gon?

Problem 4.8.6

For independent identically distributed random variables $X$ and $Y$ , show that $U = X+Y$ and $V = X-Y$ are uncorrelated but not necessarily independent. Show that $U$ and $V$ are independent if $X$ and $Y$ are $N(0,1)$ .

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Problem 4.8.7

(a)(b)(c)(d)

Let $X$ and $Y$ have a bivariate normal density with zero means, variances $\sigma^{2}, \tau^{2}$ , and correlation $\rho$ . Show that:

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(a)

$\mathbb {E}\left[X \mid Y\right] = \frac{\rho \sigma }{\tau } Y$ ,

(b)

$\operatorname {Var}\left[X \mid Y\right] = \sigma^{2}\left(1-\rho^{2}\right)$ ,

(c)

$\mathbb {E}\left[X \mid X+Y = z\right] = \frac{\left(\sigma^{2}+\rho \sigma \tau \right) z}{\sigma^{2}+2 \rho \sigma \tau +\tau^{2}}$ ,

(d)

$\operatorname {Var}\left[X \mid X+Y = z\right] = \frac{\sigma^{2} \tau^{2}\left(1-\rho^{2}\right)}{\tau^{2}+2 \rho \sigma \tau +\sigma^{2}}$ .

Problem 4.8.8

Let $X$ and $Y$ be independent $N(0,1)$ random variables, and let $Z = X+Y$ . Find the distribution and density of $Z$ given that $X > 0$ and $Y > 0$ . Show that

\mathbb {E}\left[Z \mid X > 0, Y > 0\right] = 2 \sqrt{2 / \pi }

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Problem 4.8.9

Let $X$ and $Y$ be independent $N(0,1)$ random variables. Find the joint density function of $U = a X+b Y$ and $V = b X-a Y$ , and hence show that $U$ is $N\left(0, a^{2}+b^{2}\right)$ . Deduce a proof of the last part of Example (4.8.3).

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Problem 4.8.10

Let $X$ and $Y$ have joint distribution function $F$ , with marginals $F_{X}$ and $F_{Y}$ . Show that, if $\mu = \mathbb {E}\left[X+Y\right]$ is well defined, then its value is determined by knowledge of $F_{X}$ and $F_{Y}$ .

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Problem 4.8.11

(a)(b)

Let $S = U+V+W$ be the sum of three independent random variables with the uniform distribution on $(0,1)$ .

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(a)

Show that $\mathbb {E}\left[\operatorname {Var}\left[U \mid S\right]\right] = \frac{1}{18}$ .

(b)

Find $v(s) = \operatorname {Var}\left[U \mid S = s\right]$ for $0 < s < 3$ .