MCQ

True or False

**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) |
q^{2} |
2pq | p^{2} |

then E(x) = 2p.

#### Options

True

False

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#### Solution

Since given data is p.m.f. of r.v. X, we get

q^{2} + 2pq + p^{2} = 1

∴ (q + p)^{2} = 1

∴ (q + p) = 1 ...(i)

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

= 0 x q^{2} + 1 x 2pq + 2 x p^{2}

= 2pq + 2p^{2}

= 2p (q + p)

= 2p ...[From (i)]

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

= (0)^{2} x q^{2} + (1)^{2} x 2pq + (2)^{2} x p^{2}

= 2pq + 4p^{2}

∴ Var(X) = E(X^{2}) – [E(X)]^{2}

= 2pq + 4p^{2} – (2p)^{2}

= 2pq + 4p^{2} – 4p^{2}

= **2pq is True**.

Is there an error in this question or solution?

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