Question

The random variable X can take only the values 0, 1, 2, 3. Given that P(2) = P(3) = p and P(0) = 2P(1). if `Sigmap_ix_i^2 = 2Sigmap_ix_i`, Find the value of p.

Answer 1

It is given that the random variable X can take only the values 0, 1, 2, 3.

Given:

P(0) = 2P(1) and P(2) = P(3) = p

Let P(1) = q.

Then, P(0) = 2q

Now,

P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1

⇒2q + q + p + p = 1

⇒3q = 1 − 2p

`=> q = (1 - 2p)/3` ....(1)

Since, `Sigmap_ix_i^2 = 2Sigmap_ix_i`

∴ 2q(0)² + q(1)² +p(2² + 3²) = 2[2q × 0 + q × 1 + p(2 + 3)]

⇒ q + 13p = 2(q + 5p)

⇒ q + 13p = 2q + 10p

⇒ q = 3p

`=> (1 - 2p)/3 = 3p`          [Using (1)]

⇒ 1 - 2p = 9p

⇒ 11p = 1

`=> p = 1/11`