Solving different kinds of prob questions

Question

I have started to learn Probability. I have difficulties with solving some problems. I need help how to solve these kind of questions. I have wrote $3$ of them below:

1. Moment generating function of rv $X$ is $$m_X(t)=\frac16^{−} + \frac12^{−2} + \frac13^{−3}$$ Find mean and variance of $X$.

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2. Let $X$ be a geometric rv with parameter $p$. Show that $$P( = + \mid > ) = P( = )$$

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3. The probability $p$ of being able to log on to a computer from a remote terminal at any given time is $0.7$.

• a) What is the probability that at least $5$ attempts will be made to gain access to the computer?
• b) What is the expected number of attempts that must be made to gain access to the computer? (Define random variable for the question, determine its type and solve.)

Answer 1

1. The $n$-th moment $EX^n$of a rv $X$ with moment generating function $m_X(t)$ can be calculated as $$EX^n=\frac{d^nm_X(t)}{dt^n}\Bigg|_{t=0}$$ In your case, e.g. $$EX=\frac{dm_X(t)}{dt}\Bigg |_{t=0}=\left(-\frac16e^{-t}-\frac22e^{-2t}-\frac33e^{-3t}\right)\Bigg|_{t=0}=-\frac{13}6$$ To calculate the variance, recall that $$Var(X)=EX^2-\left(EX\right)^2$$so you just need to calculate the second moment $EX^2$ as above, by differentiating one more time.
2. Since $X\sim$geometric$(p)$, you know that $P(X=k)=p(1-p)^{k-1}$ for $k\ge1$. Hence \begin{align}P(X=n+k \mid X> n)&=\frac{P(X>n\mid X=n+k)P(X=n+k)}{P(X>n)}\\[0.4cm]&=\frac{1\cdot p(1-p)^{n+k-1}}{\sum_{m=n+1}^{+\infty}p(1-p)^{m-1}}=\frac{p(1-p)^{n}(1-p)^{k-1}}{p(1-p)^n\sum_{m=0}^{+\infty}(1-p)^m}\\[0.4cm]&=\frac{(1-p)^{k-1}}{\frac1{1-(1-p)}}=p(1-p)^{k-1}=P(X=k)\end{align} By the way, this is called the memoryless property of the geometric distribution.
3. Hint: The number $X$ of times until the first success (first time to gain access) is a geometric random variable with parameter $p=0.7$. You want to find
   • a) $P(X\ge 5)$. Use 2. or that $P(X\ge 5)=1-P(X\le 4)$.
   • b) $E[X]$.