Standard Distributions: Normal, Exponential, Poisson and Chi-Square Previous Year Questions (PYQs) for GATE DA: 5+ Solved Questions with Step-by-Step Solutions

Suppose a random variable $Z$ follows $Normal(\mu =0,{\sigma }^2=1)$ distribution with probability density function $g(z)$ and cumulative distribution function $G(z)$.
Another random variable $Y$ follows $t_1$ distribution with probability density function $h(y)$ and cumulative distribution function $H(y)$. Let $c$ be the positive real number for which $g(c)=h(c)$.
Which of the following statements is/are correct?

Let $X$ be a random variable exponentially distributed with parameter $\lambda >0$. The
probability density function of X is given by:
$$f_X(x)=\{\begin{array}{ll}\lambda e^{-\lambda x},& x\ge 0\\ 0,& otherwise\end{array}$$
If $5E(X)=Var(X)$, where $E(X)$ and $Var(X)$ indicate the expectation and variance
of $X$, respectively, the value of $\lambda$ is ______ (rounded off to one decimal place).

It is given that $P(X\ge 2)=0.25$ for an exponentially distributed random variable $X$ with $E[X]=\frac{1}{\lambda }$, where $E[X]$ denotes the expectation of $X$. What is the value of $\lambda$?
(ln denotes natural logarithm)

Consider the following statements:
(i) The mean and variance of a Poisson random variable are equal.
(ii) For a standard normal random variable, the mean is zero and the
variance is one.
Which ONE of the following options is correct?