Discrete and Continuous Random Variables and Their Probability Distributions

Introduction

Random variables and their probability distributions are essential tools in probability and statistics. They are used to model real-world phenomena where outcomes are uncertain, such as games of chance, risk assessments, and predictive analytics.

Key Concepts

1. Discrete Random Variables

• Definition: A random variable that takes on a finite or countable number of distinct values.
• Probability Distribution: Specifies the probabilities of each possible value of the random variable.
• Example:

  A discrete random variable \( X \) with the following probability distribution:

  X	1	2	3	4	5
  P(X=x)	0.1	0.2	0.15	0.05	0.5

  The sum of probabilities: \( \sum P(X=x) = 1 \)

2. Continuous Random Variables

• Definition: A random variable that takes on an infinite number of values within a continuous range.
• Probability Distribution: Represented by a probability density function (PDF), where the area under the curve equals 1.
• Example: 

3. Expected Value (Mean) for Discrete Data

• Definition: The weighted average of all possible values of a random variable.
• Formula:

  \( E(X) = \sum xP(X=x) \)

• Interpretation: Represents the long-term average value of the random variable.

Solved Examples

Example 1: Calculating Expected Value

Problem: Given the probability distribution:

Find \( E(X) \).

Solution:

• Using the formula:

  \( E(X) = \sum xP(X=x) \)

• Calculations:

  \( E(X) = (1)(0.1) + (2)(0.2) + (3)(0.15) + (4)(0.05) + (5)(0.5) \)

  \( E(X) = 0.1 + 0.4 + 0.45 + 0.2 + 2.5 = 3.65 \)

Answer: \( E(X) = 3.65 \).

Example 2: Finding a Missing Probability

Problem: A random variable \( X \) has a probability distribution:

If \( \sum P(X=x) = 1 \), find \( x \).

Solution:

• Total probability:

  \( \frac{4}{18} + \frac{5}{18} + \frac{4+x}{18} = 1 \)

  \( \frac{13 + x}{18} = 1 \)

  \( 13 + x = 18 \)

  \( x = 5 \)

Answer: \( x = 5 \).

Example 3: Fair Game

Problem: A game is defined as fair if \( E(X) = 0 \). A player pays ₹10 to roll a die and receives $ 30 if the die shows 6. Is this a fair game?

Solution:

• Possible outcomes:
  • Gain \( X = $ 20 \) (if the die shows 6)
  • Loss \( X = -$ 10 \) (if the die does not show 6)
• Probabilities:
  • \( P(X = $ 20) = \frac{1}{6} \)
  • \( P(X = -$10) = \frac{5}{6} \)
• Expected value:

  \( E(X) = (20)\frac{1}{6} + (-10)\frac{5}{6} \)

  \( E(X) = \frac{20}{6} – \frac{50}{6} = -\frac{30}{6} = -5 \)

Answer: \( E(X) = -$ 5 \). The game is not fair.

Applications and Connections

• Games of Chance: Discrete random variables are used to calculate odds in games like dice, cards, and roulette.
• Economic Models: Continuous random variables model stock prices, interest rates, and consumer behavior.

Ethical Considerations:

• Aim 8: Theories based on calculable probabilities (e.g., in casinos) may be misleading when applied to complex real-world systems like economics, where probabilities are not always calculable.
• TOK: What constitutes a “fair” game? Is it ethical for casinos to profit from games designed to ensure the house always wins?