Discrete Probability Distributions: Concepts, Examples, and Applications Chapter. 4

Discrete Probability Distributions

Introduction to Discrete Probability Distributions

Discrete probability distributions are fundamental in statistics for modeling situations where outcomes are countable and distinct. This topic covers the definitions, properties, and examples of discrete random variables and their probability distributions.

• Discrete Random Variable: A variable that can take on a finite or countable number of possible values. Examples include the number of heads in coin tosses or the number of sales calls completed in a day.

• Continuous Random Variable: A variable that can take on any value within an interval, including decimals. For example, the time spent on a phone call.

• Key Difference: Discrete variables are countable (e.g., 0, 1, 2, 3), while continuous variables can take any value within a range.

Example: The number of Fortune 500 companies that increased profits last year is discrete, while the volume of gasoline in a tank is continuous.

Probability Distributions for Discrete Variables

Definition and Properties

A probability distribution for a discrete random variable lists each possible value the variable can take, along with its probability. The probabilities must satisfy:

• Each probability is between 0 and 1: $0 \leq P(x) \leq 1$

• The sum of all probabilities is 1: $\sum P(x) = 1$

Constructing a Probability Distribution: Examples

• Coin Toss Example: Flip a coin 3 times. Let $x$ be the number of heads. The possible values for $x$ are 0, 1, 2, 3. The probability distribution is:

x	P(x)
0	1/8
1	3/8
2	3/8
3	1/8

• Dice Example: Roll two 6-sided dice. Let $x$ be the sum of the numbers. The possible values for $x$ are 2 through 12, with varying probabilities.

x	P(x)
2	1/36
3	2/36
4	3/36
5	4/36
6	5/36
7	6/36
8	5/36
9	4/36
10	3/36
11	2/36
12	1/36

Additional info: The distribution of sums when rolling two dice is symmetric and approximates a normal distribution as the number of dice increases.

Interpreting Probability Distributions

• The probability of an event is the sum of the probabilities of the outcomes included in the event.

• For example, the probability of getting a sum of 7 or 11 when rolling two dice is $P(7) + P(11) = 6/36 + 2/36 = 8/36$.

Mean, Variance, and Standard Deviation of Discrete Distributions

Calculating the Mean (Expected Value)

The mean or expected value of a discrete probability distribution is calculated as:

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

• Multiply each value of $x$ by its probability $P(x)$ and sum the results.

• Example: For a personality inventory with $x$ values 1 to 5 and corresponding probabilities, the mean is:

x	P(x)	xP(x)
1	0.16	0.16
2	0.22	0.44
3	0.28	0.84
4	0.20	0.80
5	0.14	0.70

Sum: $\mu = 2.94$

Variance and Standard Deviation

The variance of a discrete probability distribution is:

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

The standard deviation is the square root of the variance:

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

• These measures describe the spread of the distribution around the mean.

Expected Value in Context

Definition and Interpretation

The expected value is the long-run average value of repetitions of the experiment it represents. In games of chance or profit/loss scenarios, it represents the average gain or loss per trial.

• Formula: $E(x) = \sum x P(x)$

• Example: In a lottery with tickets costing $2 and prizes of $500, $250, and $100, the expected value can be calculated by multiplying each outcome by its probability and summing the results.

Binomial Distributions

Definition and Properties

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• There are $n$ trials.

• Each trial has two possible outcomes: success or failure.

• The probability of success ($p$) is constant for each trial.

• The random variable $x$ counts the number of successes: $x = 0, 1, 2, ..., n$.

Binomial Probability Formula

The probability of exactly $x$ successes in $n$ trials is:

$P(x) = \binom{n}{x} p^x q^{n-x}$

where $q = 1 - p$ and $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

• Example: If a surgery has a 90% success rate and is performed on 3 patients, the probability of exactly 2 successes is:

$P(2) = \binom{3}{2} (0.9)^2 (0.1)^1 = 3 \times 0.81 \times 0.1 = 0.243$

Mean, Variance, and Standard Deviation of Binomial Distributions

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard Deviation: $\sigma = \sqrt{npq}$

Example: If 56% of days in June are cloudy in Pittsburgh ($n=30$, $p=0.56$):

• Mean: $\mu = 30 \times 0.56 = 16.8$

• Variance: $\sigma^2 = 30 \times 0.56 \times 0.44 = 7.4$

• Standard Deviation: $\sigma = \sqrt{7.4} \approx 2.7$

Geometric Distributions

Definition and Properties

A geometric distribution models the number of trials needed to get the first success in a sequence of independent Bernoulli trials (each with probability $p$ of success).

• Trials are independent.

• Probability of success is constant.

• The random variable $x$ is the trial number of the first success: $x = 1, 2, 3, ...$

The probability that the first success occurs on trial $x$ is:

$P(x) = p q^{x-1}$

where $q = 1 - p$.

Poisson Distributions

Definition and Properties

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a constant mean rate and independence between events.

• Used for modeling counts of events (e.g., number of accidents per month).

• Parameter: $\lambda$ (mean number of occurrences in the interval).

The probability of exactly $x$ occurrences is:

$P(x) = \frac{e^{-\lambda} \lambda^x}{x!}$

where $e \approx 2.71828$.

• Example: If the average number of accidents per month is 3, the probability of 4 accidents in a month is:

$P(4) = \frac{e^{-3} 3^4}{4!}$

Summary Table: Key Discrete Distributions

Distribution	Random Variable	Parameters	Probability Formula
Binomial	Number of successes in $n$ trials	$n$, $p$	$P(x) = \binom{n}{x} p^x q^{n-x}$
Geometric	Trial of first success	$p$	$P(x) = p q^{x-1}$
Poisson	Number of events in interval	$\lambda$	$P(x) = \frac{e^{-\lambda} \lambda^x}{x!}$

Conclusion

Discrete probability distributions are essential for modeling and analyzing count data in statistics. Understanding their properties, formulas, and applications allows for effective problem-solving in various real-world contexts, from games of chance to quality control and risk assessment.