Random Variables and Probability Distributions: Study Notes for Business Statistics

Random Variables and Probability Distributions

Introduction

This chapter introduces the fundamental concepts of random variables and probability distributions, which are essential for understanding uncertainty and making informed business decisions. Both discrete and continuous random variables are discussed, along with their associated probability models and key properties.

Definitions and Types of Random Variables

Random Variables

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

• Discrete Random Variable: Can take on a finite or countable number of possible values (e.g., number of defective products in a batch).

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

Example: The number of cars passing through a toll booth in an hour (discrete); the time it takes to serve a customer (continuous).

Probability Models and Expected Value

Probability Model

• The set of all possible values of a random variable and their associated probabilities.

• Applicable to both discrete and continuous random variables.

Expected Value

• The mean (average) value of a random variable, calculated when the probability model is known.

Formula:

$E(X) = \sum [x_i \cdot P(x_i)]$

Discrete Random Variables: Calculations

Calculating Expected Value, Variance, and Standard Deviation

1. Step 1: List all possible values and their probabilities.

2. Step 2: Calculate the expected value (mean): $E(X) = \sum [x_i \cdot P(x_i)]$

3. Step 3: Calculate the deviation: $x_i - E(X)$

4. Step 4: Square the deviations: $(x_i - E(X))^2$

5. Step 5: Multiply squared deviations by their probabilities: $P(x_i) \cdot (x_i - E(X))^2$

6. Step 6: Sum the results to get the variance: $Var(X) = \sum [P(x_i) \cdot (x_i - E(X))^2]$

7. Step 7: Take the square root of the variance to get the standard deviation: $SD(X) = \sqrt{Var(X)}$

Example Table:

Outcome (x)	Probability P(x)
100	0.6
200	0.3
300	0.1

Shortcuts for Calculating E(X), Var(X), and SD(X)

• Adding a Constant (c):

  • $E(X + c) = E(X) + c$

  • $Var(X + c) = Var(X)$

  • $SD(X + c) = SD(X)$

• Multiplying by a Constant (a):

  • $E(aX) = aE(X)$

  • $Var(aX) = a^2 Var(X)$

  • $SD(aX) = |a| SD(X)$

Combining Random Variables

Adding and Subtracting Random Variables

• If X and Y are independent random variables:

  • $E(X \pm Y) = E(X) \pm E(Y)$

  • $Var(X \pm Y) = Var(X) + Var(Y)$

• If X and Y are correlated, the variance formula includes the covariance term:

  • $Var(X \pm Y) = Var(X) + Var(Y) \pm 2Cov(X, Y)$

Covariance: Measures the strength of the linear association between two variables.

Types of Probability Distributions

Discrete Probability Distributions

• Uniform Distribution: All possible outcomes have the same probability.

• Bernoulli Distribution: Models a single trial with two possible outcomes (success/failure).

• Binomial Distribution: Models the number of successes in a fixed number of independent Bernoulli trials.

• Geometric Distribution: Models the number of trials until the first success.

Uniform Distribution

• Each outcome is equally likely.

• Example: Rolling a fair die.

• Variance: $Var(X) = \frac{(n^2 - 1)}{12}$ for n outcomes.

Bernoulli Trial

• Only two possible outcomes: success (probability p) and failure (probability 1-p).

• Trials are independent.

Geometric Distribution

• Probability that the first success occurs on the k-th trial: $P(X = k) = (1-p)^{k-1}p$

• Expected value: $E(X) = \frac{1}{p}$

• Standard deviation: $SD(X) = \sqrt{\frac{1-p}{p^2}}$

Binomial Distribution

• Probability of k successes in n independent trials: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Expected value: $E(X) = np$

• Variance: $Var(X) = np(1-p)$

• Standard deviation: $SD(X) = \sqrt{np(1-p)}$

Continuous Probability Distributions

Continuous Uniform Distribution

• All intervals of the same length within the distribution's range are equally probable.

• Probability density function: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

• Expected value: $E(X) = \frac{a+b}{2}$

• Variance: $Var(X) = \frac{(b-a)^2}{12}$

Normal Distribution

• Symmetric, bell-shaped, and characterized by mean ($\mu$) and standard deviation ($\sigma$).

• Standard Normal Distribution: Mean = 0, SD = 1.

• Standardization (z-score): $z = \frac{X - \mu}{\sigma}$

• Used to find probabilities and percentiles for normally distributed variables.

Example: If $X$ is normally distributed with mean 88 and SD 4, the z-score for $X = 90$ is $z = \frac{90 - 88}{4} = 0.5$.

Summary Table: Key Discrete Distributions

Distribution	Probability Formula	Expected Value	Variance
Uniform	$P(X = x) = \frac{1}{n}$	$\frac{n+1}{2}$	$\frac{(n^2 - 1)}{12}$
Bernoulli	$P(X = 1) = p$, $P(X = 0) = 1-p$	$p$	$p(1-p)$
Binomial	$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$	$np$	$np(1-p)$
Geometric	$P(X = k) = (1-p)^{k-1}p$	$\frac{1}{p}$	$\frac{1-p}{p^2}$

Applications in Business

• Probability distributions are used to model demand, forecast sales, assess risk, and make decisions under uncertainty.

• Understanding expected value and variance helps in evaluating insurance policies, investment returns, and quality control processes.

Practice Problems

• Several practice questions are provided throughout the slides, including calculations for expected value, variance, and standard deviation for different distributions.

• Problems also include applications of the binomial and normal distributions to real-world business scenarios.

Additional info: Some formulas and examples were expanded for clarity and completeness. Practice tables and step-by-step solutions were summarized for brevity.