Conditional cdf of exponential variable

Question

Given the rate parameter for an exp r.v, I am able to calculate conditional pdf and mean on the condition of A > c. For conditional pdf I calculate P(A>c) and divide that by the pdf of the r.v. I calculate conditional mean in a similar fashion. How do I go about conditional cdf? Do I integrate the conditional pdf? What are the integral limits?

Answer 1

$\checkmark $ Yes, that's right. You integrate. The lower limit will be $c$, since the condition is that $X>c$, and the upper limit will be $x$, since you wish to find: $\mathsf P(X\leq x \mid X>c)$.

Hint: use the 'dummy' variable $s$ for the integral.

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You were given the pdf $f_X(x)= \lambda e^{-\lambda x} \;\mathbf 1_{x>0}$.

You calculated the conditional pdf $f_X(x\mid X> c) = \dfrac{f_x(x) \mathbf 1_{x>c}}{\mathsf P(X>c)} = \lambda e^{\lambda(c -x)} \quad\mathbf 1_{x>c}$

You wish to determine the conditional CDF: $\displaystyle F_X(x\mid X>c) = \int_c^x f_X(s\mid X>c) \operatorname d s \quad\mathbf 1_{x>c} \\[2ex]\displaystyle \qquad\qquad\qquad= \int_c^x\lambda e^{\lambda (c-s)} \operatorname d s \quad\mathbf 1_{x>c} $

Can you do it from here?