Q. A random variate $X$ takes the values 0, 1, 2, 3 and its mean is 1.3. If $P(X = 3) = 2P(X = 1)$ and $P(X = 2) = 0.3$,then $P(X = 0)$ is equal to
A) 0.1
B) 0.2
C) 0.3
D) 0.4
Explanation
(d)
Given, mean$= \Sigma X_{k}P(X = k) = 1.3$
$\Rightarrow \ X_{0}P(X = 0) + X_{1}(X = 1) + X_{2}P(X = 2)$
$+ X_{3}P(X = 3) = 1.3$
$\Rightarrow \ 0.P(X = 0) + 1.P(X = 1) + 2.P(X = 2)$
$+ 3.P(X = 3) = 1.3$
$\Rightarrow \ P(X = 1) + 2(0.3) + 3.2P(X = 1) = 1.3$
$\Rightarrow \ 7P(X = 1) = 0.7$
$\Rightarrow \ P(X = 1) = 0.1$
Since, $P(X = 3) = 2P(X = 1) = 2(0.1) = 0.2$
Also, $P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1$
$\Rightarrow \ P(X = 0) + 0.1 + 0.3 + 0.2 = 1$
$\Rightarrow \ P(X = 0) = 1 - 0.6 = 0.4$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.