Question

The random variable X has the following probability distribution where a and b are some constants:

X	1	2	3	4	5
P(X)	0.2	a	a	0.2	b

If the mean E(X) = 3, then find values of a and b and hence determine P(X ≥ 3).

Answer 1

We have, 

X	1	2	3	4	5
P(X)	0.2	a	a	0.2	b

Given E(X) = 3

E(X) = ∑x_ip_i = 1 × 0.2 + 2 × a + 3 × a + 4 × 0.2 + 5 × b

= 0.2 + 2a + 3a + 0.8 + 5b

⇒ 3 = 5a + 5b + 1

⇒ 5a + 5b = 2           ...(i)

We know that ∑p_i = 1

⇒ 0.2 + a + a + 0.2 + b = 1

⇒ 2a + b = 0.6       ...(ii)

On solving Eqs. (i) and (ii), we get

a = 0.2 and b = 0.2

Now, P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) 

= 0.2 + 0.2 + 0.2 = 0.6