GATE | GATE 2017 Mock | Question 65

A random variable x takes values 0, 1, 2, … with probability proportional to (x+1)(1/5)x.
The probability that x <= 5 is: (A) 0.6997
(B) 0.7997
(C) 0.8997
(D) 0.9997


Answer: (D)

Explanation:

P[X=x]\propto \frac{(x+1)}{5^{x}} \\ \Rightarrow P[X=x]=\frac{k(x+1)}{5^{x}} \\ Since \sum_{x=0}^{\infty}P(X=x)=1\Rightarrow \sum_{x=0}^{\infty}\frac{k(x+1)}{5^{x}} = 1 \\ \Rightarrow k\left \{ 1+\frac{2}{5}+\frac{3}{5^{2}}+\frac{4}{5^{3}}+\frac{5}{5^{4}}+. . . \right \}=1 \\ \Rightarrow k\left \{ 1-\frac{1}{5} \right \}^{-2}=1 \\ \Rightarrow k\left \{ \frac{25}{16} \right \}=1 \\ \therefore k=\frac{16}{25} \\ so, P(X<=5) \\ =P[X=0]+P[X=1]+...+P[X=5] \\ =\left [ k+\frac{2k}{5}+\frac{3k}{5^{2}}+\frac{4k}{5^{3}}+\frac{5k}{5^{4}}+\frac{6k}{5^{5}} \right ] \\ =\left [ k+\frac{2}{5}+\frac{3}{5^{2}}+\frac{4}{5^{3}}+\frac{5}{5^{4}}+\frac{6}{5^{5}} \right ] \\ =\frac{16}{25}\left [ 1+\frac{2}{5}+\frac{3}{5^{2}}+\frac{4}{5^{3}}+\frac{5}{5^{4}}+\frac{6}{5^{5}} \right ] \\ =0.9997

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