Question

The p.m.f. of an r.v. X is ______ 

P(x) = ${\displaystyle \{\begin{array}{ll}\frac{x^2}{\text{k}}\text{;}& x=-4\text{,}-3\text{,}-2\text{,}-1\\ 0\text{;}& \text{otherwise}\end{array}}$

Then, E(X) = ______ 

Options

• ${\displaystyle -\frac{7}{3}}$

• ${\displaystyle -\frac{5}{3}}$

• ${\displaystyle -\frac{10}{3}}$

• ${\displaystyle -\frac{8}{3}}$

Answer 1

The p.m.f. of an r.v. X is ______ 

P(x) = ${\displaystyle \{\begin{array}{ll}\frac{x^2}{\text{k}}\text{;}& x=-4\text{,}-3\text{,}-2\text{,}-1\\ 0\text{;}& \text{otherwise}\end{array}}$

Then, E(X) = ${\displaystyle \underline{-\frac{10}{3}}}$

Explanation:

X = x	-4	-3	-2	-1
P(X = x)	${\displaystyle \frac{16}{k}}$	${\displaystyle \frac{9}{k}}$	${\displaystyle \frac{4}{k}}$	${\displaystyle \frac{1}{k}}$

Since, P(-4) + P(-3) + P(-2) + P(-1) = 1

∴ ${\displaystyle \frac{16}{k}+\frac{9}{k}+\frac{4}{k}+\frac{1}{k}=1}$

⇒ ${\displaystyle \frac{30}{k}=1}$

⇒ k = 30

∴ E(X) = ∑x_i.P(x_i)

= ${\displaystyle -4.\frac{16}{30}-3.\frac{9}{30}-2.\frac{4}{30}-1.\frac{1}{30}}$

= ${\displaystyle -\frac{100}{30}}$

= ${\displaystyle -\frac{10}{3}}$