Discrete Probability Distributions and Expected Value: Study Notes

Discrete Probability Distributions

Definition and Properties

A discrete probability distribution describes the probabilities of the possible values of a discrete random variable. A random variable is discrete if it can take on a countable number of distinct values.

• Discrete random variable: A variable that can take on only specific, separate values (e.g., number of children, number of sex partners).

• Continuous random variable: A variable that can take on any value within a range (e.g., height, weight).

• Properties of a discrete probability distribution:

  • All probabilities are between 0 and 1, inclusive.

  • The sum of all probabilities is 1.

Example: The probability distribution for the number of bald eagles in a country is discrete because the number of eagles is countable.

Identifying Discrete Probability Distributions

• Check that each probability is between 0 and 1.

• Check that the sum of all probabilities equals 1.

Example Table:

x	P(x)
0	0.31
10	0.17
20	0.17
30	0.14
40	0.16

Sum of probabilities: $0.31 + 0.17 + 0.17 + 0.14 + 0.16 = 0.95$ (Additional info: The sum should be 1 for a valid probability distribution; if not, check for missing values or rounding errors.)

Mean (Expected Value) of a Discrete Random Variable

Definition and Formula

The mean (or expected value) of a discrete random variable is a measure of the center of its probability distribution. It is calculated as:

$\mu = \sum x P(x)$

• $x$ = value of the random variable

• $P(x)$ = probability of $x$

Example: For the following distribution:

x	P(x)
0	0.31
10	0.17
20	0.17
30	0.14
40	0.16

Mean: $\mu = (0 \times 0.31) + (10 \times 0.17) + (20 \times 0.17) + (30 \times 0.14) + (40 \times 0.16)$

Graphing Probability Distributions

Shape and Interpretation

The graph of a discrete probability distribution typically has:

• A horizontal axis labeled with the values of the random variable (e.g., "Age" or "Number of Children").

• A vertical axis labeled "Probability" with intervals from 0 to 1.

• Vertical line segments at each value of the random variable, with heights corresponding to $P(x)$.

Shape Descriptions:

• Skewed right: Most probabilities are concentrated at lower values; tail extends to the right.

• Skewed left: Most probabilities are concentrated at higher values; tail extends to the left.

• Symmetric: Probabilities are evenly distributed around the mean.

Example Table:

Number of Children Killed	Probability
0	0.40
1	0.20
2	0.06
3	0.01
4	0.01
5	0.02
6	0.21

Median and Mean Comparison

Definitions

• Median: The value that divides the probability distribution into two equal halves.

• Mean: The expected value, as calculated above.

Comparison: In a skewed distribution, the mean is pulled toward the tail, while the median is more resistant to extreme values.

Example: If the mean is larger than the median, the distribution is likely skewed right.

Standard Deviation and Variance

Definitions and Formulas

• Variance ($\sigma^2$): Measures the spread of the distribution.

• Standard deviation ($\sigma$): The square root of the variance.

Formulas:

$\sigma^2 = \sum (x - \mu)^2 P(x)$

$\sigma = \sqrt{\sigma^2}$

Example: For a distribution with mean $\mu = 3.1$ and probabilities as above, calculate variance and standard deviation using the formulas.

Expected Value in Applications

Definition and Interpretation

The expected value is the long-run average value of repetitions of the experiment it represents. In practical terms, it is used to predict outcomes over many trials.

• Insurance: The expected value helps companies set premiums to ensure profitability over many policies.

• Business: Expected value can be used to estimate average costs or profits.

Example Table:

Program	Total Charge ($)
A	968.000
B	1060.000
C	1690.618

Expected value: $E(X) = \sum x P(x)$

Resistant Measures

Median vs. Mean

• Median: Resistant to extreme values (outliers).

• Mean: Sensitive to extreme values.

Example: In a distribution of number of sex partners, the median may be less affected by a few individuals with very high numbers than the mean.

Summary Table: Discrete vs. Continuous Random Variables

Type	Definition	Examples
Discrete	Countable number of possible outcomes	Number of children, number of eagles
Continuous	Uncountable number of possible outcomes	Height, weight

Key Formulas

• Mean (Expected Value): $\mu = \sum x P(x)$

• Variance: $\sigma^2 = \sum (x - \mu)^2 P(x)$

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

Applications and Interpretation

• Use the mean to predict average outcomes over many trials.

• Use the standard deviation to understand variability in outcomes.

• Use the median for a measure less affected by outliers.

Additional info: These notes expand on the brief question prompts by providing definitions, formulas, and context for discrete probability distributions, expected value, and their applications in statistics.