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Discrete Random Variables

How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.

Probability Density Function

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 \( a? \)

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