Probability and Random Variables

Probability Basics

Probability is a measure of how likely an event is to occur. It forms the foundation for statistical inference and decision-making under uncertainty.

• Probability is a number between 0 and 1, where 0 means an event cannot occur and 1 means it is certain to occur.

• Experiment: A process that leads to well-defined outcomes.

• Sample Space (S): The set of all possible outcomes of an experiment.

• Event: Any subset of the sample space.

Example: Tossing a fair coin. Sample space: {Heads, Tails}. Probability of Heads = 0.5.

Events

Events are collections of outcomes. Understanding how events relate helps in calculating probabilities.

• Simple Event: An event with a single outcome.

• Compound Event: An event with more than one outcome.

• Mutually Exclusive Events: Events that cannot occur at the same time.

• Complementary Events: Two events are complements if one occurs exactly when the other does not.

Example: Rolling a die. Event A: rolling an even number. Event B: rolling an odd number. A and B are mutually exclusive.

Some Rules of Probability

Rules of probability help calculate the likelihood of complex events.

• Addition Rule: For any two events A and B:

• Multiplication Rule: For independent events A and B:

• Complement Rule:

Example: Probability of drawing a red or a king from a deck of cards.

Discrete Random Variables and Probability Distribution

A discrete random variable takes on a countable number of distinct values, each with an associated probability.

• Random Variable (X): A variable whose value is a numerical outcome of a random phenomenon.

• Probability Distribution: Lists each possible value of the random variable and its probability.

Example: Number of heads in two coin tosses. Possible values: 0, 1, 2.

The Mean and Standard Deviation of a Discrete Random Variable

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

• Mean (Expected Value):

• Standard Deviation:

Example: Calculating the expected value for the number of heads in two coin tosses.

The Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Conditions: Fixed number of trials (n), two possible outcomes (success/failure), constant probability (p), independent trials.

• Probability of k successes:

• Mean:

• Standard Deviation:

Example: Probability of getting exactly 3 heads in 5 coin tosses with .

The Normal Distribution

Areas under the Standard Normal Curve

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Areas under the curve represent probabilities.

• Z-score: The number of standard deviations a value is from the mean.

• Use standard normal tables or technology to find areas (probabilities) for given z-scores.

Example: Probability that a standard normal variable is less than 1.5.

Working with Normally Distributed Variables

Many real-world variables are approximately normally distributed. Probabilities for any normal variable can be found by converting to z-scores.

• To find , convert a and b to z-scores and use the standard normal table.

Example: Heights of adult males are normally distributed with mean 70 in and standard deviation 3 in. Find the probability a randomly selected male is taller than 73 in.

The Sampling Distribution of the Sample Mean

The Mean and Standard Deviation of the Sample Mean

The sampling distribution of the sample mean describes the distribution of sample means from all possible samples of a given size from a population.

• Mean of the sample mean:

• Standard deviation of the sample mean (Standard Error):

Example: If population mean is 50 and standard deviation is 10, for samples of size 25, .

The Sampling Distribution of the Sample Mean

As sample size increases, the sampling distribution of the sample mean becomes approximately normal (Central Limit Theorem), regardless of the population's distribution.

• Central Limit Theorem: For large n, the distribution of is approximately normal, even if the population is not normal.

• Application: Allows use of normal probability methods for inference about means.

Example: For n = 30, the distribution of sample means is nearly normal.