Discrete Random Variables

The probability distribution of a discrete random variable $X$ defined in the domain $x= 0, 1 ,2$ is as follows:

$\begin{aligned} P( X= 0 ) &= 0.11 \\ P( X= 1 ) &= 0.29 \\ P( X= 2 ) &= a. \end{aligned}$ 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 $a?$

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