Notes on Probability Models and Random Variables

Probability Model

Random Variables

• A variable $X$ is a random variable (rv) if its value depends on the outcome of a random event.

• Denoted using capital letters (e.g., $X$) for random variables, and lowercase letters (e.g., $x$) for particular values.

• Examples of random variables:

  • $X$ = "The sum of the scores on two tossed dice".

  • $X$ = "The number of heads in 20 coin flips".

  • $X$ = High school Science marks of a randomly selected college applicant.

Types of Random Variables

• Categorical (Qualitative)

• Numerical (Quantitative)

  • Discrete : Can take finite distinct outcomes.

  • E.g.:

    • The number of parking lots on campus.

    • The number of people in a household.

  • Continuous : Can take any numeric value within an interval.

  • E.g.:

    • The temperature in a room.

    • The speed of a vehicle on a highway.

Probability Model for Discrete RVs

• A probability model consists of:

  1. The collection of all possible values of a random variable.

  2. The probabilities of these values occurring.

Properties of Discrete Probability Distributions

• $0 \leq P(x_i) \leq 1$

• $\sum P(x_i) = 1$

Example: Tossing Three Coins

• Let $X$ = number of heads observed when tossing three unbiased coins.

• Probability Distribution:

• Calculation of probabilities:

  • $P(X \leq 2) = P(X=0) + P(X=1) + P(X=2) = \frac{1}{8} +\frac{3}{8} +\frac{3}{8} = \frac{7}{8}$

  • $P(X > 3) = 0 $

  • $P(2 \leq X \leq 3) = P(X = 2) + P(X = 3) = \frac{3}{8} + \frac{1}{8} = \frac{1}{2}$

  • $P(0 < X \leq 2) = P(X=1) + P(X=2) = \frac{3}{8} + \frac{3}{8} = \frac{6}{8}$

Expected Value

• The expected value, $E(X)$ or population mean $\mu$ of a random variable $X$ is the average value observed if the experiment is repeated numerous times.

• For discrete random variable:
  $\mu<em>X = E(X) = \sum x</em>i P(x_i)$

• Important notes:

  1. Ensure all possible outcomes are included.

  2. Validate that the probability model is correct.

Examples:

1. Coin Example:

   • For 3 coins:
     $\mu = E(X) = 0 \cdot \frac{1}{8} + 1 \cdot \frac{3}{8} + 2 \cdot \frac{3}{8} + 3 \cdot \frac{1}{8} = \frac{12}{8} = \frac{3}{2}$

2. Biased Coin:

   • $E(W) = 0 \cdot 0.01 + 1 \cdot 0.18 + 2 \cdot 0.81 = 1.8 $

Variance and Standard Deviation

• Population Variance:
  $\sigma^2<em>X = \sum (x</em>i - \mu)^2P(x_i)$

• Standard Deviation:
  $\sigma = \sqrt{\sigma^2}$

Example:

1. Coin Toss:

   • For $X$ (number of heads from slide 6):
     $\sigma^2_X = (0 - 1.5)^2 \cdot 1/8 + (1 - 1.5)^2 \cdot 3/8 + (2 - 1.5)^2 \cdot 3/8 + (3 - 1.5)^2 \cdot 1/8$
     (Calculate the final variance and take square root to find standard deviation)

More About Means and Variances

• Constant Shift:

  • Adding a constant alters mean but not variance:
    $E(X \pm c) = E(X) \pm c$

• Scaling: Multiplying alters both mean and variance:
  $E(aX) = aE(X), \ Var(aX) = a^2Var(X)$

Examples:

1. Salary Increase by $1000:
   Overall mean will shift: $E(S) = E(X) + 1000$, Variance remains unchanged.

2. Salary Increase by 20%:
   $E(S) = 1.2E(X)$
   Variance will change according to scaling.

Two Random Variables

• Mean of sum/difference:

  • $E(X \pm Y) = E(X) \pm E(Y)$

• Variance of independent variables' sum/difference:

  • $Var(X \pm Y) = Var(X) + Var(Y)$

Combining Random Variables

• For independent normal variables, sum/difference remains normal.

Probability Examples:

1. SAT Scores:

   • $P(X > Y)$ calculated using distribution of differences.

2. Weight of Bags of Carrots:

   • apply independence rules and find $P(Y > W)$

Conclusion

• Understanding and applying probability models for discrete and continuous random variables is crucial for statistical analysis and decision making.

• Calculation of expected values, variances, and understanding the behavior of sums and differences of random variables are key concepts in probability theory.