Discrete Random Variables

Definition

A random variable \(X\) is discrete if it can take on at most a countable number of possible values.

Probability Mass Function (PMF)

For a discrete random variable \(X\), the PMF \(p(x)\) of \(X\) is defined as:

\[ p(x) = P(X = x) \]

where \(p(a) \geq 0\) for countably many values \(x\).

• Properties of PMF:

  If \(X\) can take values \(x_1, x_2, \ldots\), then:

  \[ p(x_i) \geq 0, \quad i = 1, 2, 3,\ldots \]

  and \(p(x)=0\) for all other \(x\)

  Also, the total probability sums to 1: \(\sum_{i} p(x_i) = 1\)

• Graphical Representation: The PMF can be graphed by plotting \(p(x_i)\) on the y-axis against \(x_i\) on the x-axis. In short, a histogram.

Examples of PMFs

1. Example PMF values:

   \[ p(0) = \frac{1}{4}, \quad p(1) = \frac{1}{2}, \quad p(2) = \frac{1}{4} \]

2. PMF for sum of two dice rolls (values 2 to 12) follows a known distribution with probabilities such as:

   \[ p(7) = \frac{6}{36}, \quad p(2) = \frac{1}{36}, \ldots \]

Example Problem

Given:

\[ p(i) = c \frac{\lambda^i}{i!}, \quad i=0,1,2,\ldots, \quad \lambda > 0 \]

• Find:
  1. \(P(X=0)\)
  2. \(P(X > 2)\)

Solution:

• Since probabilities sum to 1:

  \[ \sum_{i=0}^\infty p(i) = 1 \implies \sum_{i=0}^\infty c \frac{\lambda^i}{i!} = 1 \]

• Recognize the series for \(e^\lambda\):

  \[ c e^\lambda = 1 \implies c = e^{-\lambda} \]

• So the PMF is:

  \[ p(i) = e^{-\lambda} \frac{\lambda^i}{i!} \]

• \(P(X=0) = e^{-\lambda} \frac{\lambda^0}{0!} = e^{-\lambda}\)

• \(P(X > 2) = 1 - P(X \leq 2) = 1 - \sum_{i=0}^2 e^{-\lambda} \frac{\lambda^i}{i!} = 1 - e^{-\lambda}\left(1 + \lambda + \frac{\lambda^2}{2}\right)\)

Cumulative Distribution Function (CDF) for Discrete Variables

• The CDF \(F(a)\) is given by:

  \[ F(a) = P(X \leq a) = \sum_{x_i \leq a} p(x_i) \]

• If \(X\) takes values \(x_1 < x_2 < x_3 < \ldots\), then \(F\) is a step function:

  • FF is constant between points \(x_{i-1}\) and \(x_i\).
  • At each \(x_i\), \(F\) jumps by \(p(x_i)\).

Example CDF

Given PMF:

\[ p(1) = \frac{1}{4}, \quad p(2) = \frac{1}{2}, \quad p(3) = \frac{1}{8}, \quad p(4) = \frac{1}{8} \]

The CDF \(F(a)\) is:

\[ F(a) = \begin{cases} 0 & a < 1 \\ \frac{1}{4} & 1 \leq a < 2 \\ \frac{3}{4} & 2 \leq a < 3 \\ \frac{7}{8} & 3 \leq a < 4 \\ 1 & a \geq 4 \end{cases} \]

Each jump size equals the probability at that point.