Question

A random variable X takes values 0, 1, 2, 3, ..... with probability proportional to (x + 1) , then:

• A. P(X = 0) = $\frac { 16 } { 25 }$
• B. P (X ≥ 1) = $\frac { 16 } { 25 }$
• C. P(X ≥ 1) = $\frac { 7 } { 25 }$
• D. None of these

Answer 1

Answer: (A)

P(X = 0) = $\frac { 16 } { 25 }$

Answer 2

P(X = 0) + P(X = 1) + .... = 1

Given P(X = x) = k(x + 1) $\left( \frac { 1 } { 5 } \right) ^ { x }$ .

So, k $\left( 1 + 2 \cdot \frac { 1 } { 5 } + 3 \cdot \left( \frac { 1 } { 5 } \right) ^ { 2 } + 4 \cdot \left( \frac { 1 } { 5 } \right) ^ { 3 } + \ldots \cdot \right) = 1$ Ž k = $\frac { 16 } { 25 }$.

Hence, P(X = 0) = $\frac { 16 } { 25 } \cdot 1 = \frac { 16 } { 25 }$.

P(X ≥ 1) = 1 - P(X = 0) = 1 - $\frac { 16 } { 25 } = \frac { 9 } { 25 }$ .