Some Questions in probability and statistics

**prosper** · January 16th 2011, 10:31 AM

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.

**Random Variable** · January 16th 2011, 11:33 AM

$\displaystyle V(X|X>0) = E(X^2|X>0) - [E(X|X>0)]^2$

$\displaystyle E(X|X>0) = \frac{\sum_{x=1}^{\infty} xf(x)}{ P(X>0)}$

$\displaystyle E(X^{2}|X>0) = \frac{\sum_{x=1}^{\infty} x^{2}f(x)}{ P(X>0)}$

$\displaystyle f(x) = \frac{1}{x!} \frac{1}{e}$

$\displaystyle P(X>0) = 1 - P(X=0) = 1 - \frac{1}{e} = \frac{e-1}{e}$

$\displaystyle \sum_{x=1}^{\infty} xf(x) =\frac{1}{e} \sum^{\infty}_{x=1} \frac{x}{x!} = \frac{1}{e} \ e = 1$

$\displaystyle \sum_{x=1}^{\infty} x^{2}f(x) =\frac{1}{e} \sum^{\infty}_{x=1} \frac{x^{2}}{x!} = \frac{1}{e} \ 2e = 2$

so $\displaystyle E(X|X>0) = \frac{e}{e-1}$

and $\displaystyle E(X^{2}|X>0) = \frac{2e}{e-1}$

then finally $\displaystyle V(X|X>0) = \frac{2e}{e-1} - \frac{e^{2}}{(e-1)^{2}} = \frac{e(e-2)}{(e-1)^{2}}$

**prosper** · January 16th 2011, 12:02 PM

Thanks! you've really helped me
any idea about the second question?

**prosper** · January 17th 2011, 10:46 AM

1. Let X1,X2 be independent random variables with equal distribution and CDF of:
$F_X (x) = { x \over x+1}, x \ge 0$
Let $Y=X1 \cdot X2$ 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....

**matheagle** · January 17th 2011, 03:02 PM

You start by drawing.
you want to be "inside" the hyperbola xy<1, where x is x1 and y is x2.
By differentiation we have the marginals ...

$f(x)=(x+1)^{-2}$ where x>0

so the joint density is $f(x,y)= (x+1)^{-2} (y+1)^{-2}$ in the first quadrant

so you need to integrate $\int_0^{\infty}\int_0^{1/x}{dydx\over (x+1)^{2} (y+1)^{2}}$

**prosper** · January 17th 2011, 03:48 PM

Thanks.

1. The probability that a flip of a coin will get head is a continuous random variable P, with this PDF:
$f_P (p) = \left\{ \begin{array}{1 1} 2 & \quad {1 \over 4} \le p \le {3 \over 4} \\ 0 & \quad \mbox{otherwise} \\ \end{array} \right.}$
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
$P_X (x_i \theta) = c(\theta) e^{-\theta x} \quad ,x=0,1,... \quad ,\theta>0$
while $c(\theta)$ is a constant who dose not depend on x:
a. $\log \left( 1 + { n \over \sum_{i=1}^n x_i \right) }$
b. $\log \left( 1 + { \sum_{i=1}^n x_i \right) }$
c. $\log \left( n + { 1 \over \sum_{i=1}^n x_i \right) }$
d. none of the above.

I got d...

**matheagle** · January 17th 2011, 04:15 PM

you left out something
I assume you want the MLE of theta, you estimate parameters.
It looks like you must solve for c, I get

$c(\theta)= 1-e^{-\theta}$ using the geometric series

so the likelihood function is

$L= \bigl(1-e^{-\theta}\bigr)^ne^{-\theta\sum x_i}$

Take the log, differentiate, then set equal to zero, flip it over and....THE answer is (a)

**prosper** · January 17th 2011, 04:20 PM

yea.. i meant the MLE

Here is another one i solved:

Let $X~U(0,\theta)$ continuous. A confidence interval to $\theta$ is suggested based on a single observation:
the confidence level of the confidence interval is:
a. $1-\alpha$
b. $1-2\alpha$
c. $1-{\alpha \over 2}$
d. you cannot calculate the confidence level from the given data

I've got b

**matheagle** · January 17th 2011, 04:34 PM

I know, but MLE of which paramater, in this case it's obvious that it's theta
BUT you just don't ask for MLE's
It's the MLE of .....(an unknown parameter)

**prosper** · January 17th 2011, 04:46 PM

I don't quite understand how you got (a) in the MLE question, especially how you found C could you please elaborate?

I've got another one. and I've added another one to the previous post i made.
I solved this one too the answer that i got is (a).

Let W~N(0,0.5). A sensor got signal W when hasnt penetration (H0) and signal W+Theta whan has penetration (alternative H1). Theta>0 constant.

Got record signal 0.96. Should we reject H0 with Statistical significance of 5% ?
a. Yes
b. No
c. The answer independ with theta
d. Threre is no way to answer about one record.

**matheagle** · January 17th 2011, 08:29 PM

A few comments
(1) You need to create a new thread when asking a different question
(2) You do need to show some work. Just saying you think the answer is (d), gets old.
(3) You should hit the thank you link once in awhile.

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