Probability and Statistics Practical Class, Week 23

Question One §

Question §

Let $V$ be a random variable with the following probability mass function. Calculate the expectation (mean, ${\mu }_V$), variance (${\sigma }_V^2$), and standard deviation (${\sigma }_V$) of $V$.

$$p_V(v)=\{\begin{array}{ll}\frac{125}{216}& \text{at }v=-1,\\ \frac{75}{216}& \text{at }v=1,\\ \frac{15}{216}& \text{at }v=2,\\ \frac{1}{216}& \text{at }v=3,\\ 0& \text{otherwise}\end{array}$$

Solution §

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Question Two §

Question §

Let $X$ be the discrete random variable $\text{number of tails shown when two coins are thrown}$. Define two more random variables, where $X_1=\text{the number of tails shown on the first coin}$ and $X_2=\text{the number of tails shown on the second coin}$.

1. Show that $X$, $X_1$, and $X_2$ are random variables.
2. Calculate the values of the expectation and the variance of the random variables.

Solution §

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Question Three §

Question §

A variable $Z$ has the following probability density function.

$$f_Z(z)=\{\begin{array}{ll}C{e}^{-0.0001z}& z\ge 0,\\ 0& z<0\end{array}$$

1. Calculate $C$ to make $f_Z(z)$ a valid probability density function.
2. Compute the expectation value of $Z$.

Solution §

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Question Four §

Question §

Calculate the expectation and variance of…

$$f_r(t)\{\begin{array}{ll}\frac{2}{15}-\frac{2}{225}t& 0\le t\le 15,\\ 0& \text{otherwise}\end{array}$$

Solution §

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Question Five §

Question §

A variable has the following probability density function,

$$f_Y(y)=\{\begin{array}{ll}\frac{1}{4}(2y+3)& 0\le y\le 1,\\ 0& \text{otherwise}\end{array}$$

1. Compute the expectation value of $Y$ and $aY$.
2. Compute the expectation value of $Y^2$ and $(aY{)}^2$.
3. Compute the variance of $Y$ and $aY$.
4. Compute the variance of $(aY+b)$.
5. Would this result apply to other random variables?

Solution §

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