Random Variables and Probability Models – Study Notes

Random Variables and Probability Models

Introduction

This chapter explores the concept of random variables, their probability models, and the calculation of expected values and variances. It also introduces important discrete probability models, including the Uniform, Geometric, Binomial, and Poisson distributions, which are foundational for statistical inference and modeling real-world random phenomena.

Random Variables

Types of Random Variables

• Random Variable: A variable whose value is determined by the outcome of a random event.

• Discrete Random Variable: Can take on a countable number of distinct values (e.g., number of books purchased).

• Continuous Random Variable: Can take on any value within a given interval (e.g., time, weight).

The probability model for a random variable lists all possible values and their associated probabilities.

Expected Value of a Random Variable

Definition and Calculation

The expected value (mean) of a discrete random variable is the long-run average value it takes after many repetitions of the random process. It is calculated as:

$E(X) = \sum x_i P(X = x_i)$

where $x_i$ are the possible values and $P(X = x_i)$ are their probabilities.

Example: Life Insurance Policy

The probability model for a life insurance policy is shown below:

Policyholder Outcome	Payout x (cost)	Probability $P(X = x)$
Death	100,000	1/1000
Disability	50,000	2/1000
Neither	0	997/1000

The expected annual payout is calculated as:

$E(X) = 100,000 \times \frac{1}{1000} + 50,000 \times \frac{2}{1000} + 0 \times \frac{997}{1000} = 200$

Standard Deviation and Variance of a Random Variable

Definition and Calculation

The variance of a discrete random variable measures the average squared deviation from the mean:

$Var(X) = \sum (x_i - E(X))^2 P(X = x_i)$

The standard deviation is the square root of the variance:

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

Example: Life Insurance Policy (Standard Deviation)

Policyholder Outcome	Payout (cost)	Probability $P(X = x)$	Deviation $(x - E(X))$
Death	100,000	1/1000	99,800
Disability	50,000	2/1000	49,800
Neither	0	997/1000	-200

Example: Book Store Purchases

• Probabilities: 0 books (0.2), 1 book (0.4), 2 books (0.4)

• Expected value: $E(X) = 0 \times 0.2 + 1 \times 0.4 + 2 \times 0.4 = 1.2$

• Standard deviation: $SD(X) = \sqrt{(0-1.2)^2 \times 0.2 + (1-1.2)^2 \times 0.4 + (2-1.2)^2 \times 0.4}$

Properties of Expected Values and Variances

Linear Transformations

• Adding a constant $c$ to $X$:

  • $E(X + c) = E(X) + c$

  • $Var(X + c) = Var(X)$

  • $SD(X + c) = SD(X)$

• Multiplying $X$ by a constant $a$:

  • $E(aX) = aE(X)$

  • $Var(aX) = a^2 Var(X)$

  • $SD(aX) = |a| SD(X)$

Addition Rule for Random Variables

• For independent random variables $X$ and $Y$:

  • $E(X \pm Y) = E(X) \pm E(Y)$

  • $Var(X \pm Y) = Var(X) + Var(Y)$

Bernoulli Trials

Definition and Properties

• Each trial has two outcomes: success (probability $p$) and failure (probability $q = 1 - p$).

• Trials are independent.

• 10% Condition: If the sample size is less than 10% of the population, independence can be assumed.

Examples: Coin tosses, yes/no survey responses, basketball free throws.

Discrete Probability Models

Uniform Model

If $X$ has $n$ possible outcomes, each equally likely, then $X$ has a Uniform distribution $U[1, ..., n]$.

Example: Tossing a fair die ($U[1,2,3,4,5,6]$), each outcome has probability $1/6$.

Geometric Model

Predicts the number of Bernoulli trials required to achieve the first success.

• Probability: $P(X = x) = q^{x-1} p$

• Expected value: $\mu = \frac{1}{p}$

• Standard deviation: $\sigma = \sqrt{\frac{q}{p^2}}$

Example: Probability that a salesman closes his first sale on the fourth attempt (with $p = 0.8$): Use the geometric model.

Binomial Model

Predicts the number of successes in a fixed number of Bernoulli trials.

• Probability: $P(X = x) = \binom{n}{x} p^x q^{n-x}$

• Mean: $\mu = np$

• Standard deviation: $\sigma = \sqrt{npq}$

Example: Probability that a tennis player makes all 6 first serves in bounds ($p = 0.67$, $n = 6$): Use the binomial model.

Expected number in bounds: $E(X) = np = 6 \times 0.67 = 4.02$

Poisson Model

Predicts the number of events that occur over a given interval of time or space. Used for modeling counts of occurrences.

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

• Expected value: $E(X) = \lambda$

• Standard deviation: $SD(X) = \sqrt{\lambda}$

Example: Number of website purchases per minute (mean rate $\lambda = 3$): Use the Poisson model.

Uniform Model Example: Satisfaction Survey

• Probability model: Uniform (all numbers equally likely).

• Probability selected number is even: $0.5$

• Probability selected number ends in 000: $0.001$

Key Points and Cautions

• Probability models are simplifications; if the model is wrong, so are the results.

• Check for independence when using models that require it (e.g., Bernoulli trials).

• Variances of independent random variables add, but standard deviations do not.

• For discrete random variables, probability models assign a probability to each possible outcome.

Summary of Formulas

• Expected value (mean): $E(X) = \sum x_i P(X = x_i)$

• Variance: $Var(X) = \sum (x_i - E(X))^2 P(X = x_i)$

• Standard deviation: $SD(X) = \sqrt{Var(X)}$

• Linear transformations:

  • $E(X \pm c) = E(X) \pm c$

  • $E(aX) = aE(X)$

  • $Var(X \pm c) = Var(X)$

  • $Var(aX) = a^2 Var(X)$

  • $SD(aX) = |a| SD(X)$

• Addition of independent random variables:

  • $E(X \pm Y) = E(X) \pm E(Y)$

  • $Var(X \pm Y) = Var(X) + Var(Y)$

Distributions Covered

• Uniform: All outcomes equally likely.

• Geometric: Number of trials until first success.

• Binomial: Number of successes in fixed number of trials.

• Poisson: Number of events in a fixed interval.