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GATE DA 2025 Set 1 — Question 40

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MCQ+2 / -0.67MediumBernoulliDiscrete Random VariablesProbability & StatisticsBinomialCentral Limit TheoremStatistical Inference

Probability & Statistics → Discrete Random Variables → Bernoulli

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Question

A random variable XX is said to be distributed as Bernoulli(θ)Bernoulli(\theta), denoted by X∼Bernoulli(θ)X \sim Bernoulli(\theta), ifP(X=1)=θ,P(X=0)=1−θP(X = 1) = \theta, \quad P(X = 0) = 1 - \thetafor 0<θ<10 < \theta < 1. Let Y=∑i=1300XiY = \sum_{i=1}^{300} X_i, where Xi∼Bernoulli(θ),i=1,2,…,300X_i \sim Bernoulli(\theta), i = 1, 2, \dots, 300 be independent and identically distributed random variables with θ=0.25\theta = 0.25. The value of P(60≤Y≤90)P(60 \le Y \le 90), after approximation through Central Limit Theorem, is given by
(Recall that ϕ(x)=12π∫−∞xe−t22dt\phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{t^2}{2}} dt)
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