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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.
Options
True
False
Solution
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.
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