Question

the pmf for the r.v. x is as shown

Answer 1

In the graph we are given 14 terms x, with the following values:

1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 5, 3, 2, 1


i) the mean of x is:
 
$\displaystyle{\frac{1+2+3+3+3+2+1+1+2+3+5+3+2+1}{14}= \frac{32}{14}= 2.3$

ii)

to find the variance, 

1.we find the difference of each value to the mean.
2. we square each of these differences
3.we find the mean of the squared differences

as follows:
i)

1-2.3=-1.3,   2-2.3=-0.3     3-2.3=0.7     5-2.3=2.7

ii)

$(-1.3)^2=1.69\\\\(-0.3)^2=0.09\\\\(0.7)^2=0.49\\\\(2.7)^2=7.29$

iii)

we have 4-1's, 4-2's, 5-3's, and 1 5, 

so the mean of the differences squared is :


$\displaystyle{ \frac{4\cdot1.69+4\cdot0.09+5\cdot0.49+7.29}{14}= \frac{16.86}{14}= 1.2$



Answer:

mean = 2.3

variance = 1.2