Question

State whether the following is True or False :

If p.m.f. of discrete r.v. X is

x	0	1	2
P(X = x)	q2	2pq	p2

then E(x) = 2p.

पर्याय

• True

• False

Answer 1

Since given data is p.m.f. of r.v. X, we get
q² + 2pq + p² = 1
∴ (q + p)² = 1
∴ (q + p) = 1                          ...(i)

E(X) = \[\sum\limits_{x=0}^{2} x\text{P}(x)\]

= 0 x q² + 1 x 2pq + 2 x p²
= 2pq + 2p²
= 2p (q + p)
= 2p                  ...[From (i)]

E(X²) = \[\sum\limits_{x=0}^{2} x^2\text{P}(x)\]

= (0)² x q² + (1)² x 2pq + (2)² x p²
= 2pq + 4p²
∴ Var(X) = E(X²) – [E(X)]²
= 2pq + 4p² – (2p)²
= 2pq + 4p² – 4p²
= 2pq is True.