What will be the probability of p(X>x+y|X>x)

Question

How to calculate the conditional probability of a random variable which is greater than some value.

Answer 1

Using the definition of conditional probability:

$$ \Pr(X>x+y \vert X>x) = \frac{\Pr(X>x+y, \, X>x)}{\Pr(X>x)} $$

Where we can say $$ \Pr(X>x+y, \, X>x) = \Pr(X>x+y) $$ for some $y\geq 0$. Otherwise $$ \Pr(X>x+y, \, X>x) = \Pr(X>x) $$ for $y<0$.

Therefore

• for $y<0$: $$ \Pr(X>x+y \vert X>x) = 1$$

• for $y\geq0$: $$ \Pr(X>x+y \vert X>x) =\frac{\Pr(X>x+y)}{\Pr(X>x)} $$