Section 6.1

Discrete Random Variables

Distinguish between the countable and the continuous, and learn how chance determines the value of variable outcomes.

1

Random Variables

A Random Variable is a numerical measure of the outcome from a probability experiment. Its value is determined by chance and is typically denoted by letters like .

Discrete

Has either a finite or countable number of values.

Visual: Represented as specific, isolated points on a number line.
  • • Number of heads in 5 coin flips
  • • Number of students in a class

Continuous

Has infinitely many values and results from measurements.

Visual: Represented as an uninterrupted continuum on a number line.
  • • Height of a person
  • • Time it takes to complete a task
2

Discrete Probability Distributions

A probability distribution provides the possible values of a random variable and their corresponding probabilities. It can be presented as a table, graph, or formula.

Required Rules

Rule 1

Each probability must be between 0 and 1, inclusive.

Rule 2

The sum of all probabilities must equal exactly 1.

3

Graphing Distributions

Discrete Probability Histograms

Discrete probability distributions are typically graphed as histograms.

Key Insight: This approach emphasizes that discrete variables have specific, isolated outcomes rather than a continuous range. The horizontal axis represents the random variable , and the vertical axis represents the probability .
Select Scenario:

100 Coin Flips (Fair)

Let X = Number of Heads in 100 Flips. This approximates a Normal bell curve.

Loading chart...
Total Probability: P(Σx) ≈ 1
4

Mean & Expected Value

The Mean ()

The mean of a discrete random variable is the weighted average of all possible values. It is computed as:

LogicLens: Expected Value

Because the mean represents long-term expectations, it is also called the Expected Value, .

Interpretation: The mean represents the long-term average outcome if the experiment is repeated many times (the Law of Large Numbers). It does not necessarily represent the result of a single experiment.

5

Standard Deviation

The standard deviation () measures the spread or dispersion of the distribution. It tells us how far, on average, the random variable is from the mean.

Formula
It is the square root of the variance. Note that we square the difference so positive and negative deviations don't cancel out.

Common Pitfalls

Real-World Application

Insurance & Risk Management

Insurance companies use Expected Value to set premiums. They calculate the average payout per policy (probability of accident × cost) and charge more than that amount to ensure profit over thousands of customers.

Casino Games

Every casino game has a negative Expected Value for the player. The "House Edge" ensures that, by the Law of Large Numbers, the casino always wins in the long run.

Interactive Calculator

Expected Value Calculator
Value ()
Probability ()
Total Probability:

Calculation Results

Expected Value
"Long-term average"
Variance
Std Dev
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