Question

Given the following distribution,

(See Below)

What is the probability that x<2 given that x<4?

Answer 1

\[f(x)= \frac{ 2 }{ x ^{3} }\] for x > 1

f(x) = 0 otherwise

Answer 2

So what I really need is to check my answer with someone... I got an answer but not sure if I did it correctly or not.

Answer 3

\[P(X<2|X<4)=\frac{P((X<2)\wedge(X<4))}{P(X<4)}=\frac{P(X<2)}{P(X<4)}\]
where
\[P(X<k)=\int_{-\infty}^kf(x)\,dx=\int_1^k\frac{2}{x^3}\,dx=\frac{1}{x^2}\bigg|_{x=k}^1=1-\frac{1}{k^2}\]