Question

Let X be a continuous random variable with a uniform distribution on the interval open square brackets 0 comma 10 close square brackets. Find straight P open parentheses straight X less than 1 space or space straight X greater than 8 close parentheses.

Answer 1

`X` has PDF

`f_X(x)=\begin{cases}\frac1{10}&\text{for }0\le x\le10\\0&\text{otherwise}\end{cases}`

and thus CDF

`F_X(x)=\begin{cases}0&\text{for }x<0\\\frac x{10}&\text{for }0\le x\le10\\1&\text{for }x>10\end{cases}`

Because
`X` is continuous, we have

`P(X<1\text{ or }X>8)=1-P(1\le X\le8)=1-(P(X\le8)-P(X\le1))`

`P(X<1\text{ or }X>8)=1-P(X\le 8)+P(X\le1)`

`P(X<1\text{ or }X>8)=1-F_X(8)+F_X(1)`

`P(X<1\text{ or }X>8)=1-\frac8{10}+\frac1{10}=\boxed{\frac3{10}}`