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

For $x \in \mathbb{R}$, the floor function is denoted by $f(x)=\lfloor x\rfloor$ and defined as follows $\lfloor x\rfloor=k, k \leq x

where $k$ is an integer. Let $Y=\lfloor X\rfloor$, where $X$ is an exponentially distributed random variable with mean $\frac{1}{\ln 10}$, where In denotes natural logarithm. For any positive integer $l$, one can write the probability of the event $Y=l$ as follows

$$ P(Y=l)=q^l(1-q) $$

The value of $q$ is

A random experiment consists of throwing 100 fair dice, each die having six faces numbered 1 to 6 . An event $A$ represents the set of all outcomes where at least one of the dice shows a 1 . Then, $\mathrm{P}(\mathrm{A})=$

Consider a coin-toss experiment where the probability of head showing up is $p$. In the $i^{\text {th }}$ coin toss, let $X_i=1$ if head appears, and $X_i=0$ if tail appears.

Consider

$$ \hat{p}=\frac{1}{n} \sum_{i=1}^n X_i $$

where $n$ is the total number of independent coin tosses.

Which of the following statements is/are correct?