 
\lastversion = 4
\hbadness = 10000
\hfuzz = 20 in
\title {Math 323 \qquad Test 3}
\date {April 6,1994}
\noprofessor

%\answersheetfootline {I have observed the honor code
%\hskip 10pt 
%   \underbar{\hskip 3 in}}
%version(1 green),(2 yellow),(3 blue),(4 pink)

%%%begin

%begin 1
\problem The random variables $X_1$ and $X_2$  have joint density function $$\displaystyle f(x_1,x_2) =\cases
{24x_2 & if $0\leq x_2 \leq x_1 \leq 1$ and 
$x_1 + x_2 \leq 1$\cr $\ 0$ &elsewhere.\cr}$$ 

Find the marginal density function $f_2(x_2)$.

\correct $24(x_2 - 2{x_2}^2)$    
\wrong $24(x_2 -{x_2}^2)$
\wrong $24(x_2 + 2{x_2}^2)$
\wrong $24(x_2 + {x_2}^2)$
\wrong $24({1\over 2} - x_2 + {x_2}^2)$  
\perm1:bcaed
\perm2:acebd
\perm3:dbcea
\perm4:cadbe


%end 1

%begin 2  
\problem For $X_1$ and $X_2$ of problem 1, find the marginal density function
$f_1(x_1)$.
\correct $12({1\over 2} - x_1 + {x_1}^2 - |x_1 -
{1\over2}|)$  
\wrong $24(x_1 - 2{x_1}^2)$ 
\wrong $24(- x_1 + {x_1}^2)$ 
\wrong $24({1\over 2} - x_1 - {x_1}^2)$ 
\wrong $24({1\over 2} + x_1 + {x_1}^2)$ 
\perm1:cbdae
\perm2:bdaec
\perm3:daecb
\perm4:aecbd


%end 2
\pagebreak

%begin 3
\problem Let $X_1$ and $X_2$ have a uniform joint probability function given by
$$\displaystyle f(x_1,x_2)=\cases{{1\over2} &if $0\leq
x_1\leq 2$ and $0\leq x_2 \leq 1$\cr&\cr 0
&elsewhere.\cr}$$                                

Determine the probability $P(X_1 + X_2 \leq 1)$.
\correct ${1\over 4}$
\wrong ${1\over 8}$
\wrong ${1\over 3}$
\wrong ${1\over 2}$
\wrong ${3\over 8}$
\perm1:abcde
\perm2:acebd
\perm3:cbaed
\perm4:cadbe


%end 3

%begin 4
\problem Two telephone calls come independently into a switchboard, and at
random times in a fixed 1-hour period.  What is the probability that they
arrive within 10 minutes of each other?
\correct ${11\over 36}$ 
\wrong ${21\over 36}$
\wrong ${13\over 36}$
\wrong ${15\over 36}$
\wrong${25\over 36}$

\perm1:acbed
\perm2:cbeda
\perm3:bedac
\perm4:adbec


%begin 5

\problem  The standard normal variable Z has $e^{t^2\over 2}$ as its
moment-generating function.  Using this information, compute $Var(Z^2)$.

\correct 2
\wrong 3
\wrong 1
\wrong 4 
\wrong 5
\perm1:baced
\perm2:acedb
\perm3:cedba
\perm4:edbac

%end5

%begin 6


\problem Let X be normally distributed with $\mu$ = 600 and $\sigma$ = 30. 
Compute $P(X>660)$.
\correct .0228
\wrong .4772 
\wrong .9772
\wrong .5228
\wrong .0013
\perm1:cbade
\perm2:badec
\perm3:adecb
\perm4:decba


%end 6

\pagebreak



%begin 7

\problem Professor Stern just gave a mathematics test to his students.  The
scores are normally distributed with a mean of 75 and a standard deviation
of 16.  What should the lowest passing grade be so that 10\% of the class fails
the test?
\correct 55     
\wrong  60
\wrong 52         
\wrong 58
\wrong 62
\perm1:bdaec
\perm2:daecb
\perm3:aecbd
\perm4:ecbda
%end 7

%begin 8
\problem  Certain steel beams have yield strengths (measured in thousands of
pounds per square inch) which follow a Weibull distribution with $\gamma$ =
2 and $\theta$ = 4,000.  A construction project uses two such beams and calls
for a yield strength in excess of 70,000 psi. What is the probability that
both beams meet the specifications for the project?
\correct .086 
\wrong .590
\wrong .228
\wrong .387
\wrong .291 
\perm1:edcba
\perm2:daecb
\perm3:aecbd
\perm4:ecbda
%end 8



%begin9

\problem A random variable X has a beta distribution with $\alpha$ = 3 and
$\beta$ = 1.  Find $P(X>{1\over 2})$.
\correct ${7\over 8}$
\wrong ${15\over 16}$
\wrong ${5\over 8}$
\wrong ${13\over16}$
\wrong ${7\over16}$
\perm1:abcde
\perm2:bcdea
\perm3:cdeab
\perm4:deabc

%end9

%begin10
\problem Let X be exponential with density function ${1\over 2}e^{-x\over
2}$ for $x>0$. Find the moment-gemerating function of X. 
\correct $(1-2t)^{-1}$ 
\wrong $(2-t)^{-1}$
\wrong $(1-t)^{-2}$
\wrong $(2t-1)^{-1}$
\wrong $(t-2)^{-2}$
\perm1:bacde
\perm2:acdeb
\perm3:cdeba
\perm4:debac
%end 10

\pagebreak

%begin 11
\problem Illustrate the usefulness of the integration formula
$\int_0^{\infty}x^n{e^{-x\over \theta}}\,dx = \Gamma(n+1)\ \theta^{n+1}$ by
computing $E[X^3]$  where X is a gamma random variable with $\alpha$ = 2 and
$\beta$ = 4.

\correct$1536$
\wrong ${1\over 14}$
\wrong  $24576$
\wrong $7680$
\wrong ${1\over 64}$
\perm1:baecd
\perm2:aecdb
\perm3:ecdba
\perm4:cdbae
%end 11


%begin 12

\problem The joint probability function for the discrete random variables
$X_1$ and $X_2$ is 0 everywhere except for the four points (1,1), (1,2),
(2,1), and (2,2).  At (2,2), p(2,2) = ${5\over 8}$ and p(1,1) = p(1,2)=
p(2,1) = ${1\over8}$. Find the expected value
$E[{X_1}{X_2}]$ of the product
${X_1}{X_2}$.


\correct ${25\over 8}$
\wrong  ${9\over 8}$
\wrong ${28\over 8}$ 
\wrong ${17\over 8}$
\wrong ${13\over 8}$
\perm1:ecbda
\perm2:bdaec
\perm3:dabce
\perm4:dabce

%end12


%begin 13
\problem Let the joint density function of $X_1$ and $X_2$ be given by
$f(x_1,x_2) = {2x_1}$for points $(x_1,x_2)$ in the unit square and $f(x_1,x_2)
= 0$ elsewhere. Find the probability 
$$P(X_2\leq{1\over 8}\ |\ X_1\leq{1\over 4}).$$

\correct ${1\over 8}$ 
\wrong  ${1\over 4}$
\wrong  ${1\over 3}$
\wrong  ${3\over 8}$
\wrong $0$
\perm1:baecd
\perm2:cdbae
\perm3:dbaec
\perm4:ecdba

%end 13
\pagebreak

%begin 14
\problem  For $X_1$ and $X_2$ of problem 13, find
$P(X_2\leq{1\over8}\ |\ X_1={1\over 4}).$  
\correct ${1\over 8}$
\wrong  ${1\over 4}$
\wrong  ${1\over 2}$
\wrong  ${1\over16}$
\wrong  ${3\over16}$
\perm1:ecbad
\perm2:decba
\perm3:ecbad
\perm4:cbade

%end 14

%begin 15
\problem Are the random variables $X_1$ and $X_2$ of problems 13 and
14 independent or not independent?      

\correct independent
\wrong not independent
\wrong don't choose this 
\wrong don't choose this
\wrong don't choose this
\perm1:cbeda
\perm2:edacb
\perm3:dacbe
\perm4:cbeda

%end 15



%begin 16
\problem The table below gives the joint probability function for the discrete random
variables $X_1$ and $X_2$.  Use it to find $P(X_1\leq 1\ |\ X_2 = 1).$


\correct ${27\over 45}$
\wrong   ${9\over 28}$ 
\wrong   ${2\over 7}$ 
\wrong   ${3\over 14}$
\wrong   ${10\over 21}$
\perm1:cdeab 
\perm2:cdeab
\perm3:deabc
\perm4:deabc 

%end16
\end

    