Verify whether \(f(x)=\begin{cases} \frac { 2x }{ 9 } ,\quad 0\le x\le \\ 0,\quad elsewhere \end{cases}\) is a probability density function

A continuous random variable. X has the p.d.f. defined by \(f(x)=\left\{\begin{array}{l} C e^{-a x}, \quad 0<x<\infty \\ 0, \quad \text { elsewhere } \end{array}\right.\) Find the value of C if a> 0

In an entrance examination a student has to answer all the 120 questions. Each question has four options and only one option is correct. A student gets 1 mark for a correct answer and loses \(\frac{1}{2}\) mark for a wrong answer. What is the expectation of the mark scored by a student if he chooses the answer to each question at random?

In a gambling game a man wins Rs. 10 if he gets all heads or all tails and loses Rs. 5 if he gets 1 or 2 heads when 3 coins are tossed once. Find his expectation of gain.

Find the mean for the probability density function \(f(x)=\begin{cases} \frac { 1 }{ 24 } ,-12\le x\le 12 \\ 0,\quad otherwise \end{cases}\)