Geometric Random Variable probability problem

[probhelp150]

Homework Statement

X is a geometric random variable with p = 0.1. Find:
$a. F_X(5)$
$b. Pr(5 < X \leq 11)$
$c. Pr(X=7|5<X\leq11)$
$d. E(X|3<X\leq11)$
$e. E(X^2|3<X\leq11)$
$f. Var(X|3<X\leq11)$

Homework Equations

The Attempt at a Solution

Can someone check my work and help me?
a.
$F_X(5) = P(X\leq5)=1-(1-p)^x$
$=1-(0.9)^5$
$=0.40951$b.
$Pr(5<X\leq11)=F_X(11)-F_X(5)$
$=0.2766794039$c. Not too sure about this one.
$Pr(X=7) = F_X(7) - F_X(6) = 0.0531441$
$Pr(5<X\leq11)=0.2766794039$
$\frac{Pr(X=7)}{Pr(5<X<\leq11)} = \frac{0.0531441}{0.2766794039}=0.192078$

d.
$Pr(3<X\leq11) = F_X(11)-F_X(3) = 0.41518940391$
$E(X|3<X\leq11) = \sum_{x=4}^{11} \frac{x*0.1*0.9^{x-1}}{Pr(3<X\leq11)}$
$=\sum_{x=4}^{11} \frac{x*0.1*0.9^{x-1}}{0.41518940391)}$
$=\frac{0.1}{0.41518940391}\sum_{x=4}^{11} {x*0.9^{x-1}}$
$=6.9533987498770702912469710479707$

e. Don't know how to do this one.
$Pr(3<X\leq11) = F_X(11)-F_X(3) = 0.41518940391$
$E(X^2|3<X\leq11) = \sum_{x=4}^{11} \frac{x^2*0.1*0.9^{x-1}}{Pr(3<X\leq11)}$
$=\sum_{x=4}^{11} \frac{x^2*0.1*0.9^{x-1}}{0.41518940391)}$
$=\frac{0.1}{0.41518940391}\sum_{x=4}^{11} {x^2*0.9^{x-1}}$
$=53.415557495820389902397015631005$
f. Variance = $E[X^2] - E[X]^2 = 5.0658033210283859687442550652196$

 

[Orodruin]

How is an expectation value defined?

 

[probhelp150]

How is an expectation value defined?

I believe expected value is the average. I reattempted the problem. Could you help me double check?

 

[Orodruin]

I get the same thing. Note that you would typically not need to quote so many decimals, it is essentially just distracting.

 

[probhelp150]

I get the same thing. Note that you would typically not need to quote so many decimals, it is essentially just distracting.

Are my answers for a,b,c right as well? I'm only using the "full" answer because I'm doing the work on my computer (Windows Calculator) and it was easier to copy/paste the answer. I know that it should be truncated on an exam or homework.

Is there another method of obtaining E[X] and E[X^2]? Some examples I've found on the internet use Summations and others use Integrals.

 

[Orodruin]

Yes, they look fine. This is the output of my Mathematica check

Note the two equivalent ways of doing (b).