Question

Probability mass function. f(x) = (125/31)(1/5)^x

find
F(1)
F(2)
P( X less or equal to 1.5)
P (X>2)
P(1 < X less or equal to 2

Answer 1

Assuming
`f(x)` is the PMF and
`F(x)` refers to the CDF, and provided that
`x\in\{1,2,3\}` (as in your previous question), you have


`F(x)=\begin{cases}0&\text{for }x<1\\\\(25)/(31)&\text{for }1\le x<2\\\\(30)/(31)&\text{for }2\le x<3\\\\1&\text{for }x\ge3\end{cases}`

This comes from the definition of the CDF:


`F(x)=\mathbb P(X\le x)=\sum_i\mathbb P(X=i)`

Now,


`F(1)=(25)/(31)`

`F(2)=(30)/(31)`

`\mathbb P(X\le1.5)=F(1.5)=(25)/(31)`

`\mathbb P(X>2)=1-\mathbb P(X\le2)=1-F(2)=1-(30)/(31)=\frac1{31}`

`\mathbb P(1<X\le2)=\mathbb P(X=2)=f(2)=\frac5{31}`