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

$F_X(x)=\begin{cases}0&\text{for }x10\end{cases}$

Because $X$ is continuous, we have

$P(X8)=1-P(1\le X\le8)=1-(P(X\le8)-P(X\le1))$

$P(X8)=1-P(X\le 8)+P(X\le1)$

$P(X8)=1-F_X(8)+F_X(1)$

$P(X8)=1-\dfrac8{10}+\dfrac1{10}=\boxed{\dfrac3{10}}$