Some Questions in probability and statistics

1. Let X~Pois(1) what is the variance of X given x is bigger than 0?

a. (e-2)/(e-1)

b. e/(e-1)

c. 1/(e-2)

d. e(e-2)/(e-1)^2

I've tried to use the conditional variance formula but I had problems with calculating the

probability of x=k and x>0.

2. A worker calls his wife once a day. the length of each talk is independent of the others, its a random variable which distributes exponentially with a mean of 5.

One day the boss tells the worker that it is unacceptable and he has 3 more chances, if he calls his wife and talks for more than 10 minutes 3 times he will be fired.

What is the expected number of days the worker has before he gets fired?

a. 2e^3

b. 3e^3

c. 3e^(-2)

d. e^2 + e^3

The parameter of the exponential distribution is 1/5... this is easy.

but I have no idea what should I do next.

some more questions I had trubles with

1. Let X1,X2 be independent random variables with equal distribution and CDF of:

$\displaystyle F_X (x) = { x \over x+1}, x \ge 0$

Let $\displaystyle Y=X1 \cdot X2$ the probability that Y is smaller or equal to 1 is:

a. 1/4

b. 1/2

c. 3/8

d. none of the above

I don't even know how to start solving this question....

another question i ran into

Thanks.

1. The probability that a flip of a coin will get head is a continuous random variable P, with this PDF:

$\displaystyle f_P (p) = \left\{ \begin{array}{1 1} 2 & \quad {1 \over 4} \le p \le {3 \over 4} \\ 0 & \quad \mbox{otherwise} \\ \end{array} \right.}$

Let A be an event in which we got an head in the flip, what is the PDF of P given A?

another question which I solved but I'm not sure about the answer is:

2. What is the maximum estimator of

$\displaystyle P_X (x_i \theta) = c(\theta) e^{-\theta x} \quad ,x=0,1,... \quad ,\theta>0$

while $\displaystyle c(\theta)$ is a constant who dose not depend on x:

a. $\displaystyle \log \left( 1 + { n \over \sum_{i=1}^n x_i \right) }$

b. $\displaystyle \log \left( 1 + { \sum_{i=1}^n x_i \right) }$

c. $\displaystyle \log \left( n + { 1 \over \sum_{i=1}^n x_i \right) }$

d. none of the above.

I got d...