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Random Variables and Probability Distributions

Random Variables and Probability Distributions

  • Understand the concept of random variables.
  • Distinguish between discrete and continuous random variables.
  • Learn about probability mass functions (PMF) and probability density functions (PDF).
  • Calculate expected value, variance, and standard deviation.
  • Visualize probability distributions.

Definition of a Random Variable

A random variable is a function that assigns numerical values to each outcome in a sample space.

  • Discrete Random Variable: Takes countable values (e.g., number of heads in a coin flip).
  • Continuous Random Variable: Takes any value in a range (e.g., height, temperature).

Probability Mass Function (PMF) and Probability Density Function (PDF)

A probability distribution describes how probabilities are assigned to different values of a random variable.

Concept Discrete (PMF) Continuous (PDF)
Definition \( P(X = x) = f(x) \), probability of discrete values \( P(a \leq X \leq b) = \int_a^b f(x) \,dx \), probability over an interval
Probability Constraint \( \sum P(X = x) = 1 \) \( \int_{-\infty}^{\infty} f(x) \,dx = 1 \)

Expectation and Variance

The expected value (mean) of a random variable \( X \) is:

\[ E[X] = \sum x P(X = x) \quad \text{(Discrete)} \] \[ E[X] = \int x f(x) \,dx \quad \text{(Continuous)} \]

The variance measures the spread of the distribution:

\[ Var(X) = E[X^2] - (E[X])^2 \]

The standard deviation is the square root of the variance:

\[ \sigma_X = \sqrt{Var(X)} \]

Proof

By definition, variance is:

\[ Var(X) = E[(X - E[X])^2] \]

Expanding the squared term:

\[ Var(X) = E[X^2 - 2X E[X] + E[X]^2] \]

Using the linearity of expectation:

\[ Var(X) = E[X^2] - 2E[X]E[X] + E[X]^2 \]

Since \( E[X] \) is a constant:

\[ Var(X) = E[X^2] - (E[X])^2 \]

Visualization of Discrete vs Continuous Distributions

Examples

Example 1: A fair die is rolled. Define the random variable \( X \) as the outcome. Compute \( E[X] \).

Since all outcomes are equally likely:

\[ E[X] = 1\cdot \frac{1}{6} + 2\cdot \frac{1}{6} + 3\cdot \frac{1}{6} + 4\cdot \frac{1}{6} + 5\cdot \frac{1}{6} + 6\cdot \frac{1}{6} \] \[ = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 \]

Exercises

  • Question 1: A coin is flipped three times. Let \( X \) be the number of heads. Find the PMF.
  • Question 2: The lifetime of a light bulb follows an exponential distribution with \( \lambda = 0.1 \). Compute \( E[X] \).
  • Question 3: A fair die is rolled. Compute \( Var(X) \).
  • Answer 1: \( P(X=0) = \frac{1}{8}, P(X=1) = \frac{3}{8}, P(X=2) = \frac{3}{8}, P(X=3) = \frac{1}{8} \).
  • Answer 2: \( E[X] = \frac{1}{\lambda} = 10 \).
  • Answer 3: \( Var(X) = \frac{35}{12} \).

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