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## Q. 4.25

If X ∼ Exp(4), find $μ_{X}$, $\sigma^{2}_{X}$, and P(X ≤ 0.2).

## Verified Solution

We compute $μ_{X}$ and $\sigma^{2}_{X}$ from Equations (4.20) and (4.21), substituting) λ = 2. We obtain $μ_{X}$ = 0.25, $\sigma^{2}_{X}$ = 0.0625. Using Equation (4.19), we find that

$μ_{X} = \frac{1}{λ}$        (4.20)
$\sigma^{2}_{X} = \frac{1}{λ²}$        (4.21)

$F(x)= P(X\leq x) \begin{cases} 1 – e^{-\lambda x} & x\gt 0 \\ 0 & x\leq 0 \\ \end{cases}$        (4.19)

P(X ≤ 0.2) = 1 – $e^{-4(0.2)}$ = 0.5507