 
\lastversion = 4
\hbadness = 10000
\hfuzz = 20 in
\title {Math 323 \qquad Final Examination}
\date {May 5,1994}
\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 Let X be a uniform random variable on the interval 
[-3,3].  For $U = X^2 + 1$ and for $1\leq u
\leq10$, we have the density function $f_U(u)$, given by 

\correct ${1\over 6\sqrt{u-1}}$    
\wrong ${1\over 12\sqrt{u-1}}$
\wrong ${1\over 6\sqrt{u^2+1}}$
\wrong ${1\over 12 \sqrt{u^2+1}}$
\wrong ${1\over 18\sqrt{u-1}}$  
\perm1:bcaed
\perm2:acebd
\perm3:dbcea
\perm4:cadbe





%end 1

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

For the function $U = X_1X_2$ find the density
function $f_U(u)$ where $0 \leq u \leq 1$.

\correct $-ln\ u$
\wrong $-u(ln\ u)$
\wrong $-2u\;(ln\ u)$
\wrong $u-u(ln\ u)$
\wrong $u - ln\ u$
\perm1:cbdae
\perm2:bdaec
\perm3:daecb
\perm4:aecbd

%end 2


%begin 3
\problem A restaurant has found that 40\% of its customers
order chicken, 20\% order beef and 40\% order fish.  A
table of 8 customers is about to order.  What is the
probability that 2 order chicken, 3 order beef and 3
order fish?
\correct $.046$
\wrong $.027$
\wrong $.083$
\wrong $.066$
\wrong $.091$
\perm1:abcde
\perm2:cebad
\perm3:cbaed
\perm4:cadbe



%end 3
\pagebreak
%begin 4
\problem Let $X_1$ and $X_2$ be discrete random variables
whose joint probability function vanishes except for the
six values: 

\settabs 5 \columns
\vskip .5 cm
\+&p(0,0)\ =\ ${1\over 6}$  & p(1,0)\ =\ ${1\over
3}$  & p(2,0)\ =\ ${1\over 6}$  \cr
\vskip .5 cm
\+&p(0,1)\ =\ ${1\over 12}$  & p(1,1)\ =\ ${1\over 6}$  &
p(2,1)\ =\ ${1\over 12}$  \cr
\vskip .5 cm
Find the covariance cov($X_1$, $X_2$).

\correct $0$ 
\wrong $1$
\wrong ${1\over 2}$
\wrong ${1\over 3}$
\wrong${-1\over 2}$

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

%begin 5

\problem  A pair of dice is rolled 18,000  times.  Find
the probabilty that a sum of 7 occurs at least 3,000
times. Use a normal approximation to the binomial.
\correct .504
\wrong .500
\wrong .508
\wrong .496
\wrong .492
\perm1:baced
\perm2:acedb
\perm3:cedba
\perm4:edbac

%end5

%begin 6

\problem Let $X_1$ and $X_2$ be independent random
variables whose standard deviations are $\sigma_1$ and
$\sigma_2$, respectively.  What is the standard deviation
of  $3X_1 + 4X_2$?
\correct $\sqrt{9{\sigma_1}^2 + 16{\sigma_2}^2}$     
\wrong $9{\sigma_1}$ + $16{\sigma_2}$
\wrong $3\sigma_1 + 4\sigma_2$        
\wrong $\sqrt{9\sigma_1 + 16\sigma_2}$
\wrong $\sqrt{3{\sigma_1}^2 + 4{\sigma_2}^2}$
\perm1:cbade
\perm2:badec
\perm3:adecb
\perm4:decba

%end 6

\pagebreak



%begin 7
 
\problem Let $X_1$ and  $X_2$ have joint density function
$$\displaystyle f(x_1,x_2)=\cases{k{x_1} &if
$0\leq 2{x_2} \leq x_1\leq 2$ \cr&\cr 0 
&elsewhere.\cr}$$ 

For $f(x_1,x_2)$ to be a legitimate density function what
must k equal?
\correct ${3\over 4}$   
\wrong  ${3\over 8}$
\wrong ${2\over 3}$         
\wrong ${1\over 4}$
\wrong ${1\over 6}$

\perm1:bdaec
\perm2:daecb
\perm3:aecbd
\perm4:eacbd

%end 7

%begin 8

\problem For the random variables of problem 7 we
have $f_X(x_1) = {3\over 8}{x_1}^2$ and $E(X_2|X_1 = x_1)$ =
${x_1\over 4}$.  Use this information to compute $E(X_2)$.

\correct ${3\over 8}$ 
\wrong  ${2\over 3}$
\wrong  ${3\over 128}$
\wrong  ${1\over 3}$
\wrong  ${3\over 64}$

\perm1:edcba
\perm2:daecb
\perm3:aecbd
\perm4:ecbda
%end 8


%begin9

\problem Let X have density function $$\displaystyle f(x)=\cases{xe^{-{x^2\over
2}} & if $x>0$\cr&\cr0&elsewhere. \cr}$$ Let $U = X^2$.  Find $f_U(2)$.

\correct ${1\over 2}e^{-1}$ 
\wrong $2e^{-2}$
\wrong $2e^{-1}$
\wrong ${1\over 2}e^{-2}$t
\wrong ${1\over 4}e^{-2}$

\perm1:bacde
\perm2:acdeb
\perm3:cdeba
\perm4:debac

%end9
\pagebreak
%begin10
\problem Let X and Y have means 3 and 4 respectively. Say
E(XY) = 18. 

Find the covariance           cov(X,Y).

\correct 6
\wrong 20
\wrong 12
\wrong 4
\wrong 16

\perm1:baecd
\perm2:ecdab
\perm3:ecdba
\perm4:cdbae

%end 10


%begin 11
\problem A pair of dice is rolled.  Given that at least
one of the dice shows a 1, what is the probability that
the sum of the dice is at most 5?

\correct ${7\over 11}$
\wrong ${10\over 11}$
\wrong  ${7\over 12}$
\wrong  ${5\over 18}$
\wrong ${5\over 12}$

\perm1:baecd
\perm2:decab
\perm3:ecdba
\perm4:cdbae

%end 11

%begin 12

\problem Let A and B be events with P(A) = 0.2, P(B) = 0.5
and $P(A\cap B) = 0.1$. Find the probability that neither
A nor B occurs..

  
\correct 0.4
\wrong  0.3
\wrong 0.2 
\wrong 0.5
\wrong 0.6
\perm1:ecbda
\perm2:bdaec
\perm3:dabce
\perm4:dabce

%end12


%begin 13

\problem In problem 12 are A and B independent?
\correct Yes
\wrong  No
\wrong  Don't choose this answer
\wrong  Don't choose this answer 
\wrong  Don't choose this answer
\perm1:baecd
\perm2:cdbae
\perm3:dbaec
\perm4:ecdba

%end 13
\pagebreak

%begin 14

\problem Let X have density function
$$\displaystyle f(x) = \cases{{(2+x)/4} &if
$-2\leq x\leq 0$\cr & \cr ${(2\ -\ x)/4}$
& if $\ 0\leq x \leq 2$.\cr}$$   

Find $F_X(1)$.
\correct ${7\over 8}$
\wrong  ${3\over 4}$
\wrong  ${5\over 8}$
\wrong  ${11\over16}$
\wrong  ${13\over16}$
\perm1:ecbad
\perm2:decba
\perm3:ecbad
\perm4:cbade

%end 14

%begin 15
\problem Four components each have an exponential
distribution with an average life of three years.  If
all are turned on, find the probability that exactly
three will be working one year later.    

\correct .417
\wrong .301
\wrong .266 
\wrong .182
\wrong .094
\perm1:cbeda
\perm2:edacb
\perm3:dacbe
\perm4:cbeda

%end 15



%begin 16

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

Find $E(X_1{|X_2 = x_2})$. 

\correct ${{3x_2 + 2}\over {6x_2 + 3}}$
\wrong ${{3x_2 + 2}\over {2x_2 + 1}}$
\wrong ${{2x_2 + 1}\over {4x_2 + 3}}$
\wrong ${{3x_2 + 1}\over {6x_2 + 1}}$
\wrong ${{2x_2 + 3}\over{x_2 + 6}}$
\perm1:cabde
\perm2:abdec
\perm3:decab
\perm4:ecabd


%end 16
\pagebreak
%begin 17

\problem Suppose that 20\% of the membership in a club
would answer ``Yes'' to the question: ``Do you like
classical music?'' What is the probability that 6 people
(randomly selected) would have to be questioned before
getting 4 ``Yes'' responses?

\correct .010
\wrong .002
\wrong .027
\wrong .032
\wrong .080
\perm1:abcde
\perm2:bcdea
\perm3:cdeab
\perm4:deabc


%end17



%begin 18

\problem The number of dandelions growing in a lawn has a
Poisson distribution with a mean of 2 dandelions per
square yard.   Find the probability that a  section of  3
square yards will contain at least 2 dandelions.

\correct $1 - 7e^{-6}$
\wrong  $1 - e^{-6}$
\wrong  $1 - e^{-2}$ 
\wrong  $1 - 6e^{-6}$
\wrong  $1 - 3e^{-2}$
\perm1:abcde
\perm2:bcdea
\perm3:cdeab
\perm4:deabc

%end 18

%begin 19

\problem A discrete random variable X has probability
function: p(0) = .1, p(1) = .5, p(2) = .4 and p(x) = 0
elsewhere. 

Find V(X).


\correct .410
\wrong  .315
\wrong  .452
\wrong  .368
\wrong  .287
\perm1:abcde
\perm2:bcdea
\perm3:cdeab
\perm4:deabc       

%end 19
\pagebreak
%begin 20

\problem Let X be a random variable with mean $\mu = 3$ and
standard deviation $\sigma = 2$.  Use Tchebysheff's
inequality to complete the next sentence.  The
probability that X is not less than 6 units from the mean
is at most


\correct ${1\over 9}$
\wrong ${1\over 4}$ 
\wrong ${1\over 36}$ 
\wrong ${1\over 25}$ 
\wrong ${1\over 16}$
\perm1:cbeda
\perm2:edacb
\perm3:dacbe
\perm4:acbed

%end 20

%begin 21

\problem Find the value of 
$$\sum_{y=3}^{\infty}\pmatrix{y-1\cr2\cr}(.1)^4(.9)^{y-3}.$$

\correct 0.1
\wrong 1
\wrong 10
\wrong ${1\over 9}$ 
\wrong 0.9
\perm1:bedac
\perm2:cedab
\perm3:edacb
\perm4:cabed

%end 21

%begin 22

\problem Among 6 motors produced by an assembly line 2 are
defective.  Three of the six are selected for sale.  Find
the probability that none of these is defective.

\correct ${1\over 5}$
\wrong ${1\over 6}$
\wrong ${1\over 4}$
\wrong ${1\over 7}$ 
\wrong ${1\over 8}$
\perm1:cbeda
\perm2:edacb
\perm3:edacb
\perm4:cbeda

%end 22
\pagebreak
%begin 23

\problem A committee consists of 3 women and 5 men.  Two
of the members are to be chosen to form a
subcommittee.  Let X denote the number of women in the
subcommittee. Find the expected value E(X).

\correct ${21\over 28}$
\wrong ${41\over 28}$
\wrong ${18\over 28}$
\wrong ${31\over 28}$ 
\wrong ${25\over 28}$
\perm1:cbeda
\perm2:edacb
\perm3:dacbe
\perm4:acbed

%end 23

%begin 24

\problem Let X be normal with $\mu = 2$ and $\sigma = 3$.
Find $P(-1 \leq X \leq 3.5).$

\correct .5328
\wrong .1915
\wrong .3830 
\wrong .3413
\wrong .6826
\perm1:edacb
\perm2:dcbae
\perm3:acbed
\perm4:cbeda

%end 24

%begin 25.

\problem Telephone calls arrive at a switchboard at the rate
of 4 per hour. Let X denote the time in hours  elapsed
before the 5th call comes in.   Which of the following is
the distribution of X?

\correct Gamma with $\alpha = 5$ and $\beta = {1\over 4}$
\wrong Poisson with $\lambda = {4\over 5}$
\wrong Gamma with $\alpha = 5$ and $\beta = 4$ 
\wrong Beta with $\alpha = 5$ and $\beta = 4$
\wrong Beta with $\alpha = 5$ and $\beta = {1\over 4}$
\perm1:ebcad
\perm2:bcade
\perm3:bcade
\perm4:adebc

%% end 25

\end


