Let $X=$ time (in minutes) that elapses between the bell and the end of the lecture in case of a college professor. Suppose $X$ has p.d.f.

$f\left(x\right)=\left\{\begin{array}{lll}\frac{3x^2}{8}& ,& 0\le x\le 2\\ 0& ,& \mathrm{otherwise}\end{array}\right.$

What is the probability that lecture continues for at least $90$ seconds beyond the bell?

The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between $25$ and $35$ minutes with p.d.f.

$f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{10},& 25\le x\le 35\\ 0 ,& \mathrm{otherwise} \end{array}\right.$

What is the probability that preparation time exceeds $33$ minutes? Also find the c.d.f. of $X$

Let $X$ be the random variable with values $x_1,x_2,x_3,x_4$  such that $2P(X=x_1)=3P(X=x_2)=P(X=x_3)=5P(X=x_4)$

Find the probability distribution of $X$.

A random variable $\mathrm{X}$ has the following probability distribution.

Then, $\mathrm{P}(1\le \mathrm{X}\le 4)$ is

A random variable $X$ has the following probability mass function

Then the value of $\lambda$ is