Question

If the random variable x has a uniform distribution between 40 and 50, then p(35 ≤ x ≤ 45) is:

Answer 1

`\mathbb P(35\le X\le45)=\mathbb P(X\le45)-\mathbb P(X\le35)=F_X(45)-F_X(35)`

where
`F_X(x)` is the cumulative distribution function of
`X`. We have probability density given by


`f_X(x)=\begin{cases}\frac1{10}&\text{for }40\le x\le50\\\\0&\text{otherwise}\end{cases}`

which yields the CDF


`F_X(x)=\begin{cases}0&\text{for }x<40\\\\(x-40)/(10)&\text{for }40\le x<50\\\\1&\text{for }x\ge50\end{cases}`

and so


`\mathbb P(35\le X\le45)=(45-40)/(10)-0=\frac12`