Determining the sampling distribution Let the random variable $X$ represent the number of defective components in 
a lot of components. Assume that $X$ can take on four values: $0, 1, 2, 3$. The probability distribution of $X$ is shown in the table below: 
 X   |  0  |  1  |  2  | 3 
P(X) | 0.4 | 0.2 | 0.1 | 0.3 
1) Randomly pick two lots of components, what is the sampling distribution of average 
number of defective components in a lot. 
2) Find $Pr(\bar{X} > 2)$.
How do I go about approaching this problem? 

The possible values of the average number are  $0$, $1/2$, and so on up to $3$, a total of seven possible values. If the random variable $Y$ is the average number, we want to compute $\Pr(Y=y)$ for these seven possible values. 
For example the probability that $Y=1$ is the probability the sum is $2$, which can happen in three ways, $(0,2)$, $(2,0)$ and $(1,1)$. Thus
$$\Pr(Y=1)=(0.4)(0.1)+(0.1)(0.4)+(0.2)(0.2).$$
Only six more to go.
