Limit Theorems and Statistical Inference - Probability and Statistics - Data Science & Artificial Intelligence Previous Year Questions

A random variable \(X\) is said to be distributed as \(Bernoulli(\theta)\), denoted by \(X \sim Bernoulli(\theta)\), if \[P(X = 1) = \theta, \quad P(X = 0) = 1 - \theta\] for \(0 < \theta < 1\). Let \(Y = \sum_{i=1}^{300} X_i\), where \(X_i \sim Bernoulli(\theta)\), \(i = 1, 2, \dots, 300\) be independent and identically distributed random variables with \(\theta = 0.25\). The value of \(P(60 \le Y \le 90)\), after approximation through Central Limit Theorem, is given by
(Recall that \(\phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{t^2}{2}} dt\))