CF

Prob & Stats 6.1 PPT Notes

Section 6.1: Discrete Random Variables

Definition of Random Variables

  • A random variable is a numerical measure of the outcome of a probability experiment.

  • Its value is determined by chance, hence the term "random."

  • Random variables are typically denoted with a capital letter (e.g., X).

  • Individual values corresponding to the random variable are represented with lowercase letters (e.g., x).

  • Example: A mother has twins.

    • Sample space for biological sex options includes: {Boy, Girl}.

    • Let X = number of boys, thus possible values of X: 0, 1, 2.

Discrete vs Continuous Random Variables

  • The concepts discussed in Chapter 1 extend to random variables:

    • Discrete Random Variable:

    • Defined as having either a finite or countable number of values.

    • Can be plotted on a number line with space between them.

    • Continuous Random Variable:

    • Defined as having infinitely many values.

    • When plotted on a number line, the values are plotted uninterrupted.

Discrete Probability Distributions

  • The probability distribution of a discrete random variable X provides:

    • Possible values of the random variable.

    • Corresponding probabilities for these values.

  • Forms of probability distribution can be represented as:

    • Table

    • Graph

    • Formula

  • Rules to Follow (from Section 5.1):

    • The sum of all probability values must equal 1:
      ext{∑P(x) = 1}

    • All probability values must satisfy the inequality:
      0 < P(x) < 1

Examples of Valid and Invalid Probability Distributions

x

P(x)

1

0.20

2

0.35

3

0.12

4

0.40

5

-0.07

P(x)

1

0.20

2

0.25

3

0.10

4

0.14

5

0.31

Graphing Discrete Probability Distributions

  • Horizontal Axis: Represents the possible values of X.

  • Vertical Axis: Represents the probability associated with each value.

  • Once plotted, draw vertical lines to connect points.

  • The graph can be used to visualize the shape of the distribution.

Finding the Mean of a Discrete Random Variable

  • Formula for Mean:
    ext{Mean} = ext{E}(X) = ext{∑[x imes P(x)]}

  • Plain English Explanation: Multiply each value of X by its probability and sum the results.

  • The mean represents the average value of X expected over many trials.

  • Law of Large Numbers: The larger the number of trials (n), the closer the sample mean will approach the calculated mean.

Example Calculation of Mean

  • To calculate the mean for a given distribution, use:

    • StatCrunch: Navigate through the path: Stat > Calculators > Custom.

Expected Value

  • The expected value is another term for the mean of the probability distribution of X.

    • Denoted as E(X).

  • Represents the long-term average of the random variable over many trials.

  • Common contexts include insurance and gambling examples:

    • Example Scenario:

    • Investment in a raffle ticket costing $5. If 50 tickets are sold and one wins $200, calculate expected value:

      • Expected value calculation considers odds and returns on investment.

Another Example of Expected Value Calculation

  • Situation: John purchases a term life insurance policy for $350.

  • If John dies, the insurance payout is $250,000.

  • Estimated survival probability: 0.998937.

  • Calculate expected value for the insurance company using:

    • Strategy: Consider payout probability versus survival probability to determine AV and expected costs.

Standard Deviation (SD) and Variance

  • Standard deviation and variance can be calculated both manually or using StatCrunch.

  • Two mathematically equivalent formulas exist to compute SD:

    • Variance denoted as σ² (population variance).

    • You are NOT required to compute SD by hand.

Example of Standard Deviation and Variance Calculation

X

P(x)

1

0.10

2

0.30

3

0.45

4

0.15

  • Steps involve calculating mean, variance, and subsequently SD from a probability distribution.

Discussion Questions

  • Engaging questions for discussion on discrete distributions:

    • What is the definition of a random variable?

    • What key factors distinguish discrete from continuous random variables?

    • What qualifies as a valid discrete probability distribution?

    • How can one find the mean and standard deviation for a distribution?

    • What must be calculated when asked for the expected value?