Random Variables and Probability Distributions: Study Notes

Random Variables

Definition and Introduction

A random variable is a numerical measure of the outcome from a probability experiment, determined by chance. Random variables are typically denoted by letters such as X.

• Discrete random variable: Has a finite or countably infinite number of values.

• Continuous random variable: Has infinitely many values that are uncountable.

Example: Toss a coin 3 times and let X be the number of heads. Possible values for X are 0, 1, 2, and 3.

Random Variables: Visual Representation

A random variable X maps each outcome e in the sample space to a numerical value x = X(e) on the real number line.

Discrete and Continuous Random Variables

Definitions

• Discrete random variable: Values can be plotted on a number line with space between each point (e.g., 0, 1, 2, 3, 4).

• Continuous random variable: Values can be plotted on a line in an uninterrupted fashion, representing an interval or continuum.

Examples

• Discrete RVs: Number of sales, number of calls, number of people in line, number of mistakes per page.

• Continuous RVs: Length, depth, volume, time, weight.

Probability Distributions

Definition

A probability distribution for a discrete random variable X is a list of all possible values of X together with their corresponding probabilities. It provides a complete description of the random variable.

• Can be specified by tables, graphs (plots), or mathematical formulas (equations).

Probability Mass Function (pmf)

The probability mass function (pmf) of a discrete random variable X is the function f(x) that gives the probability that X takes on the value x.

• Requirements for pmf f(x):

  • $0 \leq f(x) \leq 1$ for all values of $x$

  • $\sum f(x) = 1$ where the summation is over all possible values of $x$

Probability Distribution Table

Representation of the probability distribution for a discrete random variable X:

X	x1	x2	...	xN
f(x) = P(X = x)	f(x1)	f(x2)	...	f(xN)

Example: Probability Distribution Table

The table below shows the probability distribution for the random variable X, where X represents the number of movies streamed on Netflix each month.

x	P(x)
0	0.06
1	0.58
2	0.22
3	0.10
4	0.03
5	0.01

Checking Validity of Probability Distributions

To determine if a table represents a valid probability distribution, check:

• All probabilities are between 0 and 1.

• The sum of all probabilities equals 1.

Example: If any probability is negative or the sum is not 1, it is not a valid probability distribution.

Probability Histogram

Definition

A probability histogram is a histogram where the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value.

Example: The probability histogram for the number of movies streamed on Netflix each month visually displays the distribution from the table above.

Mean of Discrete Random Variable

Definition and Formula

The mean (expected value) of a discrete random variable X is given by:

$\mu_X = \sum [x \cdot P(x)]$

where x is the value of the random variable and P(x) is the probability of observing x.

• The mean is not necessarily a possible value of the random variable; it represents the average behavior.

Examples

• Single die: What is the mean of X where X denotes the outcome of throwing a fair six-sided die?

• Two dice: What is the mean of X where X denotes the sum of outcomes of two fair six-sided dice?

Interpretation of Mean

If an experiment is repeated n independent times and the value of the random variable X is recorded, the mean value of the n trials will approach $\mu_X$ as n increases.

$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}$

The difference between $\bar{x}$ and $\mu_X$ gets closer to 0 as n increases.

Interpretation of Mean: Example

A basketball player shoots three free throws 100 times. The mean number of free throws made is:

$\bar{x} = \frac{3 + 2 + 3 + \cdots + 3}{100} = 2.35$

As the number of repetitions increases, the sample mean approaches the expected value.

Mean as an Expected Value

Definition

The mean of a random variable represents what we would expect to happen in the long run. It is also called the expected value, $E(X)$, of the random variable.

Example: Expected Value in Insurance

Suppose an 18-year-old male buys a $250,000 1-year term life insurance policy for $350. The probability of surviving the year is 0.999; the probability of dying is 0.001.

x	P(x)
$350$ (survives)	0.999
$-249,650$ (dies)	0.001

The expected value for the insurance company is:

$E(X) = \mu_X = \sum [x \cdot P(x)] = 350(0.999) + (-249,650)(0.001) = 100$

The company expects to make $100 for each 18-year-old male client it insures, as a long-term average.

Variance and Standard Deviation

Definitions and Formulas

• Variance of a discrete random variable X is:

$\sigma^2 = Var(X) = E[(X - \mu)^2] = \sum (x - \mu)^2 f(x)$

• Shortcut formula for variance:

$\sigma^2 = \sum x^2 f(x) - \mu^2$

• Standard deviation (SD): The positive square root of the variance.

$\sigma = \sqrt{\sigma^2}$

Mean and Variance for Random Variables: Examples

• Probability distribution of X: Mean = 2.5, $\sigma^2 = 0.81$

• Probability distribution of Y: If $Y = 3X$, then:

  • Mean of Y is 3 times the mean of X

  • Variance of Y is 9 times the variance of X (since variance scales with the square of the constant)

Rules for Variances

Variance of the Sum of Two Variables

In general, the variance of the sum of two random variables is not simply the sum of their variances.

• Example: Let X be the number of heads and Y the number of tails in 4 tosses of a fair coin. Then X + Y = 4 always, so $\sigma^2_{X+Y} = 0$.

• In general:

$Var(X + Y) \neq Var(X) + Var(Y)$

Additional info: If X and Y are independent, then $Var(X + Y) = Var(X) + Var(Y)$, but if they are dependent, this does not hold.