Let $Y=Z^2, Z=\frac{X-\mu}{\sigma}$, where $X$ is a normal random variable with mean $\mu$ and variance $\sigma^2$. The variance of $Y$ is

Let $A \in \mathbb{R}^{n \times n}$ be such that $A^3=A$. Which one of the following statements is ALWAYS correct?

Consider the cumulative distribution function (CDF) of a random variable X :

$$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq-1 \\ \frac{1}{4}(x+1)^2 & -1 \leq x \leq 1 \\ 1 & x \geq 1 \end{array}\right. $$

The value of $P\left(X^2 \leq 0.25\right)$

A random variable X is said to be distributed as $\operatorname{Bernoulli}(\theta)$, denoted by $X \sim \operatorname{Bernoulli}(\theta)$, if

$$ P(X=1)=\theta, P(X=0)=1-\theta $$

for $0<\theta<1$. Let $Y=\sum_{i=1}^{300} X_i$. Where $X_i \sim \operatorname{Bernoulli}(\theta), i=1,2, \ldots \ldots, 300$ be independent and identically distributed random variables with $\theta=0.25$. The value of $P(60 \leq \mathrm{Y} \leq 90)$, after approximation through Central Limit Theorem, is given by

(Recall that $\phi(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{t^2}{2}} d t$ )