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For a Poisson distributed random variable X, P(X = 4) = P(X = 5). What is the mean and variance of the distribution.
If X is a random variable with E(X) = 3 and E(X^2) = 16, find a lower bound for the probability P[-2 < X < 8).

The manufacturer of the ColorSmart-5000 television set claims 95 percent of its sets last at least five years without needing a single repair. In order to test this claim, a consumer group randomly selects 424 consumers who have owned a ColorSmart-5000 television set for five years. Of these 424 consumers, 326 say their ColorSmart-5000 television sets did not need a repair, whereas 98 say their ColorSmart-5000 television sets did need at least one repair.
If a random variable U has the I-distribution with n degrees of freedom, show tbat
Z = U' has the F-distribution with I and n degrees of freedom. (S)

Obtain the maximum likelihood estimate of 0 in

f(x,9) •(l +9)x1

0 < x < 1
Prove that

.A.,( m,n )-- .Jm.Jn

for all m > O,n > O.Hence evaluate 13(10,7) and P(�·iJ (6)

) Let X ,X , ••• ,X. be a random sample from a normal population, N(µ,cr ).0btain
1 2 2
a (I -a) level confidence interval for µ when cr

is unknown
) Let the p.d.f. ofa continuous random variable X be given by

f(x)=Pe P1, x�O

Find the maximum likelihood estimator of p based on a sample X" X2 , •••• , x..
) Let X and Y have the continuous joint distribution with p.d.f.

l 1- if x2 + y

f(x,y)= na''

2 :Sa'
0 , otherwise

where a> 0. Compute the marginal densities of X and Y. Check whether X and Y are independent.
6 observations on (X, Y)yielded the following data:

rxi = so, rvi = rso, rxivi = 1000.

rxf =200, rv,2 =5642.

i) Determine the correlation coefficient between X and Y.
ii) Given X =IO, what will be the predicated value ofY?

iii) Given Y =IS, predict X.

JfX is a random variable with E(X) = 3 and E(X2

= 16, find a lower bound for the

probability P[-2 < X < 8).