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

$F_X(x)=\begin{cases}0&\text{for }x$

and so

$\mathbb P(35\le X\le45)=\dfrac{45-40}{10}-0=\dfrac12$