Question

 If the probability mass function of a random variable is given by 𝑃(𝑋 = 𝑟) = 𝑘𝑟 3 , 𝑟 = 1,2,3, 𝑓, 𝑓𝑖𝑛𝑑 (i)the value of k, (ii) 𝑃( 1 2 < 𝑋 < 5 2 /𝑋 > 1 3 ), (iii) the mean and variance of X.

Answer 1

Given function,

$P(X=r)=kr^3, r=1,2,3$ .

The probability Distribution is given as-

(i) As we know sum of probabilitied=1

$\Rightarrow k+8k+27k=1\Rightarrow 36k=1\Rightarrow k=0.028$

(ii)$P( \dfrac{(1.2<X<5.2)}{X>1.3})=\dfrac{0.224+0.756}{0.224+0.756}=\dfrac{0.98}{0.98}=1$

(iii) Mean $=\sum XP(X)=2.744$

Variance $=\sum X^2P(X)=7.728$