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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.

# Discrete Random Variables - Probability Density Function (PDF)

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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