Let X Be the Number of Packages Being Mailed by a Randomly

Essay

Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of X is as follows: $$\begin{array} { c c c c c } \hline x & 1 & 2 & 3 & 4 \\\hline p ( x ) & .4 & .3 & .2 & .1 \\\hline\end{array}$$
a. Consider a random sample of size n = 2 (two customers), and let $\bar { X }$
be the sample mean number of packages shipped. Obtain the probability distribution of $\bar { X }$
.
b. Refer to part (a) and calculate $P ( \bar { X } \leq 2.5 )$
c. Again consider a random sample of size n = 2, but now focus on the statistic R = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of R. [Hint: Calculate the value of R for each outcome and use the probabilities from part (a).]
d. If a random sample of size n = 4 is selected, what is $P ( \bar { X } \leq 1.5 )$
? (Hint: You should not have to list all possible outcomes, only those for which $\bar { x } \leq 1.5 . )$