⧼exchistory⧽
21 exercise(s) shown, 0 hidden
SORT:
BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] and [math]Y[/math] be independent real-valued random variables

with density functions [math]f_X(x)[/math] and [math]f_Y(y)[/math], respectively. Show that the density function of the sum [math]X + Y[/math] is the convolution of the functions [math]f_X(x)[/math] and [math]f_Y(y)[/math]. Hint: Let [math]\bar X[/math] be the joint random variable [math](X, Y)[/math]. Then the joint density function of [math]\bar X[/math] is [math]f_X(x)f_Y(y)[/math], since [math]X[/math] and [math]Y[/math] are independent. Now compute the probability that [math]X+Y \le z[/math], by integrating the joint density function over the appropriate region in the plane. This gives the cumulative distribution function of [math]Z[/math]. Now differentiate this function with respect to [math]z[/math] to obtain the density function of [math]z[/math].

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] and [math]Y[/math] be independent random variables defined

on the space [math]\Omega[/math], with density functions [math]f_X[/math] and [math]f_Y[/math], respectively. Suppose that [math]Z = X + Y[/math]. Find the density [math]f_Z[/math] of [math]Z[/math] if

  • [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $-1 \leq x \leq +1,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
  • [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $3 \leq x \leq 5,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
  • [[math]] f_X(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $-1 \leq x \leq 1,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
    \smallskip
    [[math]] f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \;\mbox{if $3 \leq x \leq 5,$} \\ 0, & \;\mbox{otherwise.} \end{array} \right. [[/math]]
  • What can you say about the set [math]E = \{\,z : f_Z(z) \gt 0\,\}[/math] in each case?
BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose again that [math]Z = X + Y[/math]. Find [math]f_Z[/math] if

  • [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} x/2, & \mbox{if $0 \lt x \lt 2,$} \\ 0, & \mbox{otherwise}. \end{array} \right. [[/math]]
  • [[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} (1/2)(x - 3), & \mbox{if $3 \lt x \lt 5,$} \\ 0, & \mbox{otherwise}. \end{array} \right. [[/math]]
  • [[math]] f_X(x) = \left \{ \begin{array}{ll} 1/2, & \mbox{if $0 \lt x \lt 2,$} \\ 0, & \mbox{otherwise}, \end{array} \right. [[/math]]
    \smallskip
    [[math]] f_Y(x) = \left \{ \begin{array}{ll} x/2, & \mbox{if $0 \lt x \lt 2,$} \\ 0, & \mbox{otherwise}. \end{array} \right. [[/math]]
  • What can you say about the set [math]E = \{\,z : f_Z(z) \gt 0\,\}[/math] in each case?
BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math], [math]Y[/math], and [math]Z[/math] be independent random variables

with

[[math]] f_X(x) = f_Y(x) = f_Z(x) = \left \{ \begin{array}{ll} 1, & \mbox{if $0 \lt x \lt 1,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]

Suppose that [math]W = X + Y + Z[/math]. Find [math]f_W[/math] directly, and compare your answer with that given by the formula in Example. Hint: See Example.

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that [math]X[/math] and [math]Y[/math] are independent and [math]Z = X + Y[/math]. Find

[math]f_Z[/math] if

  • [[math]] f_X(x) = \left \{ \begin{array}{ll} \lambda e^{-\lambda x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
    \smallskip
    [[math]] f_Y(x) = \left \{ \begin{array}{ll} \mu e^{-\mu x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
  • [[math]] \ \ \ f_X(x) = \left \{ \begin{array}{ll} \lambda e^{-\lambda x}, & \mbox{if $x \gt 0,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
    \smallskip
    [[math]] f_Y(x) = \left \{ \begin{array}{ll} 1, & \mbox{if $0 \lt x \lt 1,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]
BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose again that [math]Z = X + Y[/math]. Find [math]f_Z[/math] if

[[math]] \begin{eqnarray*} f_X(x) &=& \frac 1{\sqrt{2\pi}\sigma_1} e^{-(x - \mu_1)^2/2\sigma_1^2} \\ f_Y(x) &=& \frac 1{\sqrt{2\pi}\sigma_2} e^{-(x - \mu_2)^2/2\sigma_2^2}\ . \end{eqnarray*} [[/math]]

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that [math]R^2 = X^2 + Y^2[/math]. Find [math]f_{R^2}[/math] and [math]f_R[/math] if

[[math]] \begin{eqnarray*} f_X(x) &=& \frac 1{\sqrt{2\pi}\sigma_1} e^{-(x - \mu_1)^2/2\sigma_1^2} \\ f_Y(x) &=& \frac 1{\sqrt{2\pi}\sigma_2} e^{-(x - \mu_2)^2/2\sigma_2^2}\ . \end{eqnarray*} [[/math]]

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Suppose that [math]R^2 = X^2 + Y^2[/math]. Find [math]f_{R^2}[/math] and [math]f_R[/math] if

[[math]] f_X(x) = f_Y(x) = \left \{ \begin{array}{ll} 1/2, & \mbox{if $-1 \leq x \leq 1,$} \\ 0, & \mbox{otherwise.} \end{array} \right. [[/math]]

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Assume that the service time for a customer at a bank is

exponentially distributed with mean service time 2 minutes. Let [math]X[/math] be the total service time for 10 customers. Estimate the probability that [math]X \gt 22[/math] minutes.

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X_1[/math], [math]X_2[/math], \dots, [math]X_n[/math] be [math]n[/math] independent random

variables each of which has an exponential density with mean [math]\mu[/math]. Let [math]M[/math] be the minimum value of the [math]X_j[/math]. Show that the density for [math]M[/math] is exponential with mean [math]\mu/n[/math]. Hint: Use cumulative distribution functions.