Chapter 13: Random Variables – Discrete Probability Models, Binomial & Poisson Distributions

Random Variables

Definition and Notation

A random variable is a variable whose value is determined by the outcome of a random event. Random variables are typically denoted by capital letters (e.g., X), while specific values they take are denoted by lowercase letters (e.g., x).

• Discrete random variables can take on a finite or countable number of distinct values (e.g., number of heads in coin tosses).

• Continuous random variables can take any value within a range (e.g., weight of a dog).

Probability Models for Random Variables

Probability Distribution

A probability model (or probability distribution) for a random variable consists of:

• The set of all possible values the random variable can take.

• The probability associated with each value.

For discrete random variables, the probabilities must sum to 1, and each probability must be between 0 and 1.

Example: Insurance Payouts

The following table shows a probability model for the random variable X, representing the payout by an insurance company:

Policyholder Outcome	Payout x	Probability P(X = x)
Death	10,000	1/1000
Disability	5,000	2/1000
Neither	0	997/1000

Probabilities as decimals:

• 1/1000 = 0.001

• 2/1000 = 0.002

• 997/1000 = 0.997

Expected Value (Mean) of a Discrete Random Variable

Definition and Calculation

The expected value (or mean) of a discrete random variable X, denoted as $\mu = E(X)$, is the long-run average value of X over many repetitions of the experiment. It is calculated as:

$\mu = E(X) = \sum x \cdot P(x)$

For the insurance example:

Interpretation: On average, the insurance company expects to pay out $20 per policyholder.

Variance and Standard Deviation of a Discrete Random Variable

Definition and Calculation

The variance of a random variable X measures the spread of its possible values around the mean. The standard deviation is the square root of the variance.

Formulas:

$\text{Var}(X) = \sum (x - \mu)^2 P(x)$

$\sigma = \text{SD}(X) = \sqrt{\text{Var}(X)}$

For the insurance example, the deviations from the mean are:

Policyholder Outcome	Payout x	Probability P(X = x)	Deviation (x - μ)
Death	10,000	1/1000	9,980
Disability	5,000	2/1000	4,980
Neither	0	997/1000	-20

Discrete Probability Models: Binomial and Poisson

Binomial Model

The binomial model describes the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials (each with the same probability of success p).

• Parameters: n (number of trials), p (probability of success)

• Probability of exactly x successes:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

Mean and standard deviation:

$\mu = np$

$\sigma = \sqrt{np(1-p)}$

Poisson Model

The Poisson model is used for modeling the number of occurrences of an event in a fixed interval of time or space, especially when events are rare and the number of trials is large.

• Parameter: $\lambda$ (mean number of occurrences)

• Probability of exactly x occurrences:

$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$

Mean and variance:

$E(X) = \lambda, \quad \text{Var}(X) = \lambda, \quad \text{SD}(X) = \sqrt{\lambda}$

Summary Table: Key Properties of Discrete Probability Models

Model	Parameters	Mean	Variance	Probability Formula
Binomial	n, p	np	np(1-p)	$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$
Poisson	$\lambda$	$\lambda$	$\lambda$	$P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$

Key Takeaways

• Random variables can be discrete or continuous; their probability models describe the likelihood of each outcome.

• The expected value (mean) and standard deviation summarize the center and spread of a random variable's distribution.

• Binomial and Poisson models are fundamental for modeling counts of events under different conditions.