Random Variables

Definition and Types

A random variable is a function that assigns a real (quantitative) value to each outcome of a random experiment. The value it takes depends on the outcome of a random event.

• Discrete random variable: Takes isolated values, such as the face value of a die or the number of heads in coin tosses. All possible outcomes can be listed.

• Continuous random variable: Takes values within an interval, such as height, weight, or time taken to finish a job.

Notation: Capital letters (e.g., X, Y, Z) denote random variables; lowercase letters (e.g., x, y, z) denote specific values they assume.

Example

An engineer selects two resistors from six (three labeled 10Ω, three labeled 20Ω) to create a resistance of 30Ω. The actual resistances vary, and the sum of the selected resistances is assigned as the value of random variable X. The possible values of X are 28, 29, 30, 31, and 32.

Probability Mass Function (pmf)

Definition and Properties

The probability mass function (pmf) of a discrete random variable X is defined as:

• The pmf must satisfy:

Example Table: PMF for Resistor Example

Expectation (Expected Value)

Definition and Calculation

The expected value of a random variable X, denoted or , is the mean value expected from the probability distribution.

• For a discrete random variable:

Example Table: PMF for Recalibration

Expected number of recalibrations:

Expectation of a Function of X

For any function of a discrete random variable X:

Special cases:

• (for constant c)

Second moment:

Variance and Standard Deviation

Definition and Calculation

Variance measures the spread of a random variable's possible values. For a discrete random variable X:

Standard deviation is the positive square root of variance:

Example Table: Variance Calculation for Recalibration

Variance:

Standard deviation:

Alternative Formula for Variance

• It can be shown that

• Equivalently,

Properties of Means and Variances

Shifting and Rescaling

• Adding/subtracting a constant shifts the mean but does not change the variance or standard deviation.

• Multiplying/dividing by a constant rescales both the mean and the variance.

Formulas:

• Shifting: ,

• Rescaling: ,

• Standard deviation: ,

• Flipping:

Studying More Than One Random Variable

Linearity of Expectation

• For constants a, b, c:

Variance of Sums and Differences

• For independent random variables X and Y:

• For constants a, b, c: (if X and Y are independent)

Why does ?

• Variance measures spread; the difference of two variables can have a wider range than either alone.

Covariance and Correlation

• Covariance measures how two random variables vary together:

• For independent X and Y, (but the converse is not always true).

• Correlation standardizes covariance:

In general,

Bernoulli Trial

Definition

A Bernoulli trial is a random experiment with two possible outcomes: 1 (success) and 0 (failure). The probability of success is p, and the probability of failure is .

Binomial Model

Definition and Properties

A Binomial random variable X is the total number of successes in n independent and identically distributed Bernoulli trials. Denoted , where n is the number of trials and p is the probability of success.

• Possible values:

• Expected value:

• Variance:

Binomial Probability Formula

The probability of exactly k successes in n trials:

Where is the binomial coefficient:

Example Table: Binomial Model Applications

Worked Examples

• Industrial audit: 12 invoices sampled, 20% receive discount. Probability that fewer than 2 receive discount: Use binomial formula with n = 12, p = 0.2, k = 0 or 1.

• Grocery store: 24 packages, 5% unsealed. Calculate probabilities for none, at least one, or at most two unsealed packages using binomial probabilities.

Practice Problems

• Construct probability models for games or experiments.

• Calculate expected values and variances for given pmfs.

• Apply binomial model to real-world scenarios.

Additional info: These notes cover foundational concepts in probability and statistics, including random variables, expectation, variance, and the binomial model, which are essential for understanding statistical inference and probability distributions.