Discrete Random Variables
The probability distribution of a discrete random variable \( X \) defined in the domain \( x= 0, 1 ,2 \) is as follows:
\[\begin{align} P( X= 0 ) &= 0.11 \\ P( X= 1 ) &= 0.29 \\ P( X= 2 ) &= a. \end{align} \] Find the value of \( a. \)
What is the expectation of the discrete random variable \( X \) having the following probability density function? \[ P(X = x) = \begin{cases} \frac{x}{210} &\quad ( x = 0,1,2, \cdots 20 ) \\ 0 &\quad \text{(otherwise)} \end{cases} \]
What is the variance of the discrete random variable \( X \) having the following probability density function? \[ P(X = x) = \begin{cases} \frac{x}{120} &\quad ( x = 0,1,2, \cdots 15 ) \\ 0 &\quad \text{(otherwise)} \end{cases} \]
If the probability distribution of a discrete random variable \( X \) is given by \[ P(X=n) = 9 \left( \frac{1}{a} \right) ^n (n \ge 1), \] what is the value of \( a? \)
If the probability distribution of a discrete random variable \( X \) is given by \[ P(X=n) = 2 \left( \frac{1}{a} \right) ^n (n \ge 1), \] what is the value of \( a? \)