Discrete Random Variables and Probability Distributions: Study Notes

Discrete Random Variables

Introduction to Random Variables & Probability Distributions

In statistics, a random variable is a numerical value assigned to each outcome of a random experiment. Random variables can be classified as either discrete or continuous, depending on whether their possible values are countable or uncountable.

• Discrete Random Variable (DRV): Can take on a finite or countably infinite set of values. Example: Number of cars sold per day.

• Continuous Random Variable (CRV): Can take on any value within a given range. Example: Height of students in a class.

• Probability Distribution: Describes the probabilities associated with each possible value of a random variable.

Example Table: Criteria for a probability distribution.

Cars sold per Day (X)	Probability (P(X))
0	0.2
1	0.5
2	0.3

All probabilities must be between 0 and 1, and the sum of all probabilities must equal 1.

Probability Distribution Example:

Outcome	Probability
Win random raffle	0.01
Lose random raffle	0.99

General formula for a probability distribution:

$orall x, 0 \\leq P(X=x) \\leq 1$ $\sum P(X=x) = 1$

Application Example: Calculating the probability of at least breaking even in a lottery scenario.

• Mean Probability: $Z=0.6$

• Probability of at least breaking even: $60\%$

Identifying Discrete Random Variables

Discrete random variables are those that can be counted, such as the number of items or events. Examples include:

• The time it takes for a randomly selected runner to complete a 5K race (not discrete).

• The number of defective lightbulbs from a randomly chosen batch in a factory (discrete).

• The number of marbles drawn from a bag (discrete).

Survey Example: Number of sodas people drink per day.

Sodas per Day	Probability
0	0.10
1	0.31
2	0.29
3	0.18
4	0.12

• Probability of at least 2 sodas per day: $0.29 + 0.18 + 0.12 = 0.59$ (59%)

• Probability of between 1 and 4 sodas per day: $0.31 + 0.29 + 0.18 + 0.12 = 0.90$ (90%)

Mean (Expected Value) of Random Variables

Calculating the Mean (Expected Value)

The mean or expected value $E(X)$ of a discrete random variable is calculated by multiplying each possible value by its probability and summing the results.

Formula:

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

Example Table: Probability distribution for number of kids per household.

Kids per Household (X)	Probability (P(X))
0	0.2
1	0.5
2	0.3

Calculation:

$E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 0 + 0.5 + 0.6 = 1.1$

Practice Table: Probability distribution for number of defective lightbulbs.

Defective Bulbs (X)	Probability (P(X))
0	0.20
1	0.30
2	0.25
3	0.18
4	0.07

Expected value calculation:

$E(X) = 0 \times 0.20 + 1 \times 0.30 + 2 \times 0.25 + 3 \times 0.18 + 4 \times 0.07$

Variance & Standard Deviation of Discrete Random Variables

Calculating Variance and Standard Deviation

The variance and standard deviation measure the spread of a probability distribution. For a discrete random variable, variance is calculated as:

$\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$

Standard deviation is the square root of variance:

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

Example Table: Probability distribution for number of kids per household.

Kids per Household (X)	Probability (P(X))
0	0.2
1	0.5
2	0.3

Calculation steps:

• Find the mean $\mu$ as above.

• For each value, compute $(x_i - \mu)^2 P(x_i)$ and sum.

Practice Table: Probability distribution for number of computers repaired.

Computers Repaired (X)	Probability (P(X))
0	0.10
1	0.30
2	0.40
3	0.20

Variance formula:

$\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$

Summary Table: Properties of Discrete Random Variables

Property	Definition	Formula
Mean (Expected Value)	Average value of the random variable	$E(X) = \sum x_i P(x_i)$
Variance	Average squared deviation from the mean	$\sigma^2 = \sum (x_i - \mu)^2 P(x_i)$
Standard Deviation	Square root of variance	$\sigma = \sqrt{\sigma^2}$

Additional info: These notes cover key concepts from Chapter 5 (Probability in Our Daily Lives) and Chapter 7 (Sampling Distributions) as well as foundational material for understanding probability distributions and random variables in statistics.