Practice section 2 of 7

The discrete random variable $Y$ has distribution

Find the mode and median of $Y$.

The discrete random variable $X$ has distribution

Find the mean of $X$.

A salesperson for a medical device company has two appointments. He believes the first has a 70% chance of a deal paying 1,000 dollars commission, and the second a 40% chance of a deal paying 1,500 dollars. Assume the appointment results are independent. What is his expected commission based on these beliefs?

Let $X$ be the number of cars passing through a car wash between 4:00 pm and 5:00 pm on a sunny Friday, with probability distribution

The manager pays the attendant $Y=2X-1$ dollars. Find the attendant’s expected earnings for this time period.

Three parts from a production line are sampled and tested. Let $X$ count the defective parts, with distribution

Find the mean and variance of $X$.

The discrete random variable $X$ has probability function

$P(X=x)=\begin{cases}k(2-x),&x=0,1,2,\\k(x-2),&x=3,4,\\0,&\text{otherwise},\end{cases}$

where $k>0$. (i) Show that $k=1/6$ and state $E(X)$. (ii) Find $E(|X-2|)$. (iii) Find $\operatorname{Var}(X)$.