Question

Let X be Exponential variable with mean 1/λ. (a) Find E[X|X < c] using a calculation from conditional density of X|X < c. [You can use the fact that: integral from 0 to c of Axe^(−Ax)dx = −ce^(−Ac) + 1/A (1 − e^(−Ac)).] (b) Find another way to calculate E[X|X < c] using the memoryless property

Answer 1

f(x|X < c) = f(x)/P( X < c)

now

$P(X < c) = \int _0^c\lambda e^{-\lambda x}dx=-e^{-\lambda c}+1$

hence

$f(x|X <c) = \frac{\lambda e^{-\lambda x}}{-e^{-\lambda c}+1}$ 0 < x < c

-tr ce-tc d.r l (-e-lc +1)

where l = $\lambda$