Probability Model

Module 4 - Section 4: Probability Model

Instructor: Rosana Fok

Random Variables

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

• Notation:

  • Denoted by a capital letter, e.g., X.

  • A particular value of a random variable is denoted with a lowercase letter, e.g., x.

• Examples:

  • X = number of observed "Tail" while tossing a coin 10 times.

  • X = recovery time after a specific surgery of a randomly selected patient.

  • X = high school Math mark for a randomly selected university applicant.

Types of Random Variables

• Random variables can be categorized as:

  • Categorical

  • Quantitative

  • Quantitative random variables can be further classified into:

    • Discrete

    • Continuous

• Implications: The models for random variables depend on their type (categorical vs quantitative); models for continuous random variables will differ from discrete random variables.

Discrete Random Variables

• Definition: Discrete random variables can take one of a finite number of distinct outcomes.

• Examples:

  • The number of stores in a shopping mall.

  • The number of cars owned by a family.

  • The number of luggages each traveler carries in the airport.

• Continuous Random Variables:

  • Continuous random variables can take any numeric value within an interval of values.

  • Examples of continuous random variables:

  • Cost of books this term.

  • Height of hockey players.

Probability Model for Discrete Random Variables

• Definition: A probability model for a random variable consists of:

  • The collection of all possible values of the random variable.

  • The probabilities that the values occur.

• Properties of Discrete Probability Distributions:

  • For any outcome x_i:

  • $0 \leq P(x_i) \leq 1$

  • $\sum P(x_i) = 1$

• Table Representation:

  • Value of X: x₁, x₂, x₃, …, x_n

  • Probability: P(x₁), P(x₂), P(x₃), …, P(x_n)

Example: Tossing Coins

Unbiased Coins

• 
  

• Experiment: Toss two unbiased coins, let X equal the number of heads observed.

• 
  

• Simple Events:
  

  Coin 1	Coin 2	X	P(X)
  H	H	2	1/4
  H	T	1	1/2
  T	H	1	1/2
  T	T	0	1/4
  Result: Find P(X < 1).
  Distribution for X (Number of Heads Observed):
  X	P(X)
  ---	------
  0	1/4
  1	1/2
  2	1/4

  Example: Tossing Biased Coins

  • 
    

  • Experiment: Toss two biased coins with the chance of observing a head equals 0.7.

  • 
    

  • Random Variables:

    • Let X = 1 if the first coin is a head, X = 0 if it is a tail.

    • Let Y = 1 if the second coin is a head, Y = 0 if it is a tail.

    • Let T = X + Y.

  • 
    

  • Distribution for T:
    

    T	P(T)
    0	0.09
    1	0.42
    2	0.49

    Expected Value: Center

    • Definition: The expected value E(X) or population mean µ (mu) of a random variable X is the value you would expect to observe on average if the experiment is repeated over and over again.

    • Relationship: It is also the center of the distribution.

    • Formula for Expected Value:
      $\mu<em>X = \mu = E(X) = \sum x</em>i P(x_i)$

    • Calculation Steps:

      • Sum the products of each possible outcome and the probability that it occurs.

      • Ensure to include all possible outcomes and to verify that a valid probability model is used.

    ---

    Example: Expected Value of Coin Toss

    Unbiased Coins

    • 
      

    • From Previous Distribution:
      

      X	P(X)
      0	1/4
      1	1/2
      2	1/4
      Expected Value Calculation:
      $\mu = E(X) = 0(1/4) + 1(1/2) + 2(1/4) = 1$

      Example: Expected Value of Biased Coins

      • 
        

      • From Previous Distribution of T:
        

        T	P(T)
        0	0.09
        1	0.42
        2	0.49
        Expected Value Calculation:
        $\mu = E(T) = 0(0.09) + 1(0.42) + 2(0.49) = 1.98$

        Standard Deviation: Spread

        • Definition: Let X be a discrete random variable with probability distribution P(X).

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

        • Population Standard Deviation: The standard deviation s (sigma) of a random variable X is the square root of its variance.
          $s = \sqrt{Var(X)}$

        ---

        Example: Variance and Standard Deviation

        Coin Toss

        • 
          

        • Distribution:
          

          X	P(X)
          0	1/4
          1	1/2
          2	1/4
          Calculate population variance and standard deviation of X = number of heads observed.

          Example: Rolling a Die

          • Scenario: Tossing an unbiased die and recording the number on the upper face Y.

          • Expected Value Calculation:

          • $\mu = E(Y) = \sum y<em>i P(y</em>i) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5$

          • Variance Calculation:

          • $s^2 = (1 - 3.5)^2(1/6) + (2 - 3.5)^2(1/6) + (3 - 3.5)^2(1/6) + (4 - 3.5)^2(1/6) + (5 - 3.5)^2(1/6) + (6 - 3.5)^2(1/6) = 2.91666$

          • Standard Deviation:

            • $s = \sqrt{2.91666} = 1.7078$

          ---

          More About Means and Variances

          • Adding/Subtracting Constants:

            • Adding or subtracting a constant from data shifts the mean but does not change the variance or standard deviation.

            • $E(X \pm c) = E(X) \pm c$

            • $Var(X \pm c) = Var(X)$

          • Multiplying by Constants:

            • Multiplying each value of a random variable by a constant affects the mean and variance.

            • $E(aX) = aE(X)$

            • $Var(aX) = a^2Var(X)$

          ---

          Example: Salary Changes in a Company

          • Scenario: Monthly income for employees at ABC Company is $5830 with a standard deviation of $8000.

          • Fixed Increase:

            1. $5000 Increase:

            • The new mean:
              $E(S) = 5830 + 5000 = 10830$

            • Variance remains unchanged:
              $Var(S) = Var(X) = 8000^2$

            • Standard deviation remains unchanged:
              $SD(S) = 8000$

            1. 10% Increase:

            • New mean:
              $E(S) = 1.1(5830) = 6413$

            • New variance:
              $Var(S) = 1.1^2Var(X) = 1.1^2(8000^2)$

            • New standard deviation:
              $SD(S) = 8800$

          ---

          Two Discrete or Continuous Random Variables

          • Mean of Sum/Difference:

            • The mean of the sum (or difference) of two random variables is the sum (or difference) of the means.

            • $E(X \pm Y) = E(X) \pm E(Y)$

          • Independence and Variance:

            • If the random variables are independent, the variance of their sum (or difference) is the sum of their variances.

            • $Var(X \pm Y) = Var(X) + Var(Y)$

          ---

          Combining Random Variables (The Good News)

          • Continuity: Nearly everything stated about discrete random variables also applies to continuous random variables.

          • Normality: When independent continuous random variables have Normal models, then their sum or difference also follows a Normal distribution.

          ---

          Example: Cake Baking and Decorating

          • Background: Amy and Sharon are siblings.

            • Amy averages 62 minutes to bake a cake with a standard deviation of 2 minutes.

            • Sharon averages 66 minutes to decorate a cake with a standard deviation of 4 minutes.

          • Variables:

            • Let X_A = baking time (Amy).

            • Let X_S = decorating time (Sharon).

            • Let T = total time = X_A + X_S.

          • Expected Time:

            • Calculate expected time:

            • $E(T) = E(X<em>A) + E(X</em>S) = 62 + 66 = 128$

          • Standard Deviation:

            • Standard deviation:

            • $SD(T) = \sqrt{2^2 + 4^2} = \sqrt{20} = 4.472$

          ---

          Example: Probability of Time Exceeding a Limit

          • Question: If T is normally distributed, what is the probability that it will take Amy and Sharon more than 137 minutes to make a decorated cake?

            • Use Z-score to find:

            • P(T > 137) where $E(T) = 128$ and $SD(T) = 4.472$.

          ---

          Example: Redo Coin Toss

          • Goal: Use the concept of combining variables to find the population mean, variance, and standard deviation.

          • Distribution:

            • Repeat the calculations from earlier for the number of heads observed when tossing two fair coins.

          ---

          Example: Milk Production

          • Scenario: Two factories producing milk, Factory A and Factory B.

            • Factory A: Fill follows a normal distribution with mean 1.01 L and standard deviation 0.01 L.

            • Factory B: Fill follows a normal distribution with mean 1 L and standard deviation 0.01 L.

          • Probability Calculation:

            • Determine the probability that a randomly chosen carton of milk from Factory A has a higher amount of milk than from Factory B.

            • Let X be the fill of milk by Factory A, and Y be the fill of milk by Factory B.

            • Thus,

            • P(X > Y) = P(X - Y > 0)

            • The random variable $X - Y$ has a normal distribution with:

              • Mean: $E(X - Y) = 1.01 - 1.00 = 0.01$

              • Standard Deviation: $SD(X - Y) = \sqrt{0.01^2 + 0.01^2} = 0.01414214$

            • Final Probability:

              • $P(Z \geq k) = P(Z \geq -0.7123) = 0.7611$

          ---

          Conclusion

          • Thank you for watching the video!