Question

If X is a random variable with probability distribution as given below:

X	0	1	2	3
P(X)	2a	a	4a	3a

Then, the variance of X is ______ 

Options

• ${\displaystyle \frac{25}{29}}$

• ${\displaystyle \frac{23}{7}}$

• ${\displaystyle \frac{24}{27}}$

• ${\displaystyle \frac{29}{25}}$

Answer 1

${\displaystyle \underline{\frac{29}{25}}}$

Explanation:

The sum of all the probabilities in a probability distribution is always unity.

∴ 2a + a + 4a + 3a = 1

⇒ 10a = 1

⇒ a = ${\displaystyle \frac{1}{10}}$

E(X) = ∑x_i.P(x_i)

= ${\displaystyle 0\left(\frac{2}{10}\right)+1\left(\frac{1}{10}\right)+2\left(\frac{4}{10}\right)+3\left(\frac{3}{10}\right)}$

= ${\displaystyle \frac{9}{5}}$

Var(X) = E(X²) - [E(X)]² 

= ${\displaystyle 0^2\left(\frac{2}{10}\right)+1^2\left(\frac{1}{10}\right)+2^2\left(\frac{4}{10}\right)+3^2\left(\frac{3}{10}\right)-{\left(\frac{9}{5}\right)}^2}$

= ${\displaystyle \frac{29}{25}}$