Discrete Probability Distributions

Discrete Probability Distributions Study Notes

3. Random Experiment and Sample Space (Review)

• Definition: A random experiment is a process that generates experimental outcomes with the following properties:

  1. The experimental outcomes are well-defined and may also be listed prior to conducting the experiment.

  2. On any single repetition (trial) of the experiment, one and only one of the possible experimental outcomes will occur.

  3. The experimental outcome that occurs on any trial is determined solely by chance.

• Terminology:

  • An experimental outcome is also referred to as a sample point.

  • The sample space (denoted as $S$) for an experiment is the set of all experimental outcomes.

• Example: In the random experiment of tossing a coin:

  • Experimental outcomes are Head and Tail.

  • Sample space: $S = ext{Head, Tail}$

4. Discrete Random Variables

• Definition: A random variable is a numerical description of the outcome of an experiment.

• Discrete Random Variables: May assume either a finite number of values or an infinite sequence of values.

4.1 Examples of Discrete Random Variables

• Random Experiment: Flip a coin

  • Random Variable (x): Face of coin showing (1 if heads; 0 if tails)

• Random Experiment: Roll a die

  • Random Variable (x): Number of dots showing on top of die (1, 2, 3, 4, 5, 6)

• Random Experiment: Contact five customers

  • Random Variable (x): Number of customers who place an order (0, 1, 2, 3, 4, 5)

• Random Experiment: Operate a health care clinic for one day

  • Random Variable (x): Number of patients who arrive (0, 1, 2, 3, …)

• Random Experiment: Offer a customer the choice of two products

  • Random Variable (x): Product chosen by customer (0 if none; 1 if product A chosen; 2 if product B chosen)

4.2 Continuous Random Variables

• Definition: A continuous random variable may assume any numerical value in an interval or collection of intervals.

4.2 Examples of Continuous Random Variables

• Random Experiment: Customer visits a web page

  • Random Variable (x): Time customer spends on web page in minutes ($x ≥ 0$)

• Random Experiment: Fill a soft drink can (max capacity = 12.1 ounces)

  • Random Variable (x): Number of ounces (0 ≤ $x$ ≤ 12.1)

• Random Experiment: Test a new chemical process

  • Random Variable (x): Temperature when the desired reaction takes place (min temperature = 150°F; max temperature = 212°F, 150 ≤ $x$ ≤ 212)

• Random Experiment: Invest $10,000 in the stock market

  • Random Variable (x): Value of investment after one year ($x ≥ 0$)

5. Developing Discrete Probability Distributions

• Definition: A probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

• Probability Function: Defines the probability distribution by providing the probability for each value of the random variable, denoted by $f(x)$.

• Required Conditions for Discrete Probability Function:

  1. $f(x) ≥ 0$ (probabilities must be non-negative)

  2. $ext{Sum of all probabilities} <br>ightarrow extstyle orall x: ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } extstyle ext{ } 1$

5.1 Example: DiCarlo Motors Automobiles Sold

• The data: Number of automobiles sold during a day at DiCarlo Motors over 300 days:

  • # of Automobiles sold (x) | # of days

  • 0 | 54

  • 1 | 117

  • 2 | 72

  • 3 | 42

  • 4 | 12

  • 5 | 3

  • Total: 300

• Random Variable of interest: $x$ = number of automobiles sold during a day

5.2 Calculate the Discrete Probability Distribution,$f(x)$

• Relative frequency method to develop the discrete probability distribution:

  • # of Automobiles sold (x) | # of days | Probability f(x)

  • 0 | 54 | $f(0) = rac{54}{300} = 0.18$

  • 1 | 117 | $f(1) = rac{117}{300} = 0.39$

  • 2 | 72 | $f(2) = rac{72}{300} = 0.24$

  • 3 | 42 | $f(3) = rac{42}{300} = 0.14$

  • 4 | 12 | $f(4) = rac{12}{300} = 0.04$

  • 5 | 3 | $f(5) = rac{3}{300} = 0.01$

  • Total: $ext{Total probability} = 1.00$ by summing the individual probabilities.

6. Expected Value (μ)

• Definition: The expected value (or mean) of a random variable is a measure of its central location.

• Calculation: The expected value is calculated as a weighted average of the values the random variable may take. The weights are the probabilities:
  $E(x) = extstyle ext{Sum} extstyle ext{ } (x f(x))$

6.1 Example: DiCarlo Motors Expected Value Calculation

• Given the table:

  • $f(x) <br>ightarrow 0.18, 0.39, 0.24, 0.14, 0.04, 0.01$

• Calculation steps and respective products:

  • $x * f(x)$

  1. 0 | $0(0.18) = 0.00$

  2. 1 | $1(0.39) = 0.39$

  3. 2 | $2(0.24) = 0.48$

  4. 3 | $3(0.14) = 0.42$

  5. 4 | $4(0.04) = 0.16$

  6. 5 | $5(0.01) = 0.05$

• Total Expected Value:

  • $E(x) = extstyle ext{Sum} extstyle ext{ } (x f(x)) = 0 + 0.39 + 0.48 + 0.42 + 0.16 + 0.05 = 1.50$

• Interpretation: Over time, DiCarlo can anticipate selling an average of 1.50 automobiles per day, or 45 automobiles per month (30 * 1.50).

7. Variance and Standard Deviation

• Variance: Summarizes the variability in the values of a random variable.

• Variance of a discrete random variable:
  $Var(x) = extstyle ext{Sum} extstyle ext{ } ( (x - ext{mean})^{2} f(x) )$

• Standard Deviation: Square root of variance
  $ext{Standard Deviation} ext{(} ext{σ} ext{)} = ext{√Var(x)}$

7.1 Example Calculation for Variance and Standard Deviation

• Calculation table:

  • x | # of days | f(x) | xf(x) | x − μ | (x − μ)² | (x − μ)²f(x)

  • 0 | 54 | 0.18 | 0.00 | $0 - 1.50 = -1.50$ | $2.25$ | $0.405$

  • 1 | 117 | 0.39 | 0.39 | $1 - 1.50 = -0.5$ | $0.25$ | $0.0975$

  • 2 | 72 | 0.24 | 0.48 | $2 - 1.50 = 0.5$ | $0.25$ | $0.06$

  • 3 | 42 | 0.14 | 0.42 | $3 - 1.50 = 1.5$ | $2.25$ | $0.315$

  • 4 | 12 | 0.04 | 0.16 | $4 - 1.50 = 2.5$ | $6.25$ | $0.25$

  • 5 | 3 | 0.01 | 0.05 | $5 - 1.50 = 3.5$ | $12.25$ | $0.1225$

  • Total: Sum relevant columns to complete variance calculations.

• Expected Total: $E(x)= 1.5, Var(x)=1.25$, $ext{Standard Deviation} (σ)= 1.118$ automobiles.

8. Bivariate Probability Distributions

• Definition: A probability distribution involving two random variables.

• Examples of Bivariate Experiments:

  • Rolling a pair of dice and tossing a coin.

  • Recording percentage gains for a stock fund and a bond fund over a period of time.

• Application: Bivariate probability distributions are utilized in the construction and analysis of financial portfolios.

8.1 Financial Applications of Bivariate Probability Distributions

• Definition: A financial advisor considers a portfolio for the coming year. The probability distributions are defined for:

  • $x = ext{percent return for investing in a large-cap stock fund}$

  • $y = ext{percent return for investing in a long-term government bond fund}$

• Bivariate Probability Distribution Example:

  • Economic Scenario | Probability f(x, y) | Large-Cap Stock Fund (x) | Long-Term Government Bond Fund (y)

  • Recession | 0.10 | -40 | 30

  • Weak Growth | 0.25 | 5 | 5

  • Stable Growth | 0.50 | 15 | 4

  • Strong Growth | 0.15 | 30 | 2

8.2 Expected Return Calculation for Individual Funds

• Expected return for:

  • Stock fund: $E(x) = extstyle ext{Sum}( x * f(x, y) )$

  • $E(x) = -40 imes 0.10 + 5 imes 0.25 + 15 imes 0.50 + 30 imes 0.15 = 9.25$

  • Bond fund: $E(y)$ to be determined similarly.

8.3 Variance and Covariance Calculations

• Individual Variances: Recommendations from financial analysts include considering the standard deviation as a measure of investment risk.

  • $Var(x) = σ^2 = extstyle ext{Sum}((x - E(x))^2 f(x, y))$

  • $Var(y) = σ^2 = extstyle ext{Sum}((y - E(y))^2 f(x, y))$

• Covariance: Calculation for variables $x$ and $y$ can determine correlation:

  • $Cov(x, y) = σ_{xy} = extstyle ext{Sum} extstyle ext{ } x imes y imes (x - E(x))(y - E(y)) f(x, y)$

  • Interpretation of covariance values:

    • $Cov(x, y) > 0$: $x$ and $y$ positively correlated

    • $Cov(x, y) < 0$: $x$ and $y$ inversely correlated

    • $Cov(x, y) = 0$: $x$ and $y$ are independent

9. The Financial Portfolio Example

• Final Interpretation involves weighing the returns and risks of both assets based on investment choices.

• The decision can rely heavily on an investor's risk tolerance:

  • An aggressive investor may opt for the stock fund due to its higher expected returns.

  • A conservative investor might prefer the bond fund for its lower associated risk.

  • Alternatively, constructing a portfolio using a weighted combination of stock and bond funds is also an option.

9.1 Properties of Linear Combination of x and y

• Definition: A random variable return, $r$, defined as a linear combination of $x$ and $y$:

  • $r = ax + by$ where $a$ and $b$ are coefficients fulfilling: $a, b > 0$, $a + b = 1$.

• Example for 50% in each fund:

  • $r = 0.5x + 0.5y$

9.2 Expected Value and Variance of Linear Combination

• Expected Value Calculation:

  • $E(r) = Ea + Eb,$ where $E(r) = 0.5E(x) + 0.5E(y)$ and results in a target value for returns.

• Variance Formula of a linear combination of two random variables:

  • $Var(r) = Var(ax) + Var(by) + 2abσ<em>{xy}$ where $σ{xy}$ is the covariance of $x$ and $y$.

9.3 Financial Portfolio Conclusions

• Analyzing the properties of these computations can lead to more informed financial decisions. Expected return, variance, and risk assessments ultimately guide investment choices and strategies for risk mitigation.

9.4 Financial Strategy Recommendations

• Specific variation in portfolio allocation based on changing economic scenarios can provide a hedge against loss.

• Investment choices can be diversified by exploring various ratios to tailor portfolios to specific risk tolerances.

End of Study Notes for ECO 380 Chapter 5.