Probability Distributions: Concepts, Calculations, and Applications

Probability Distributions

Introduction to Probability Distributions

Probability distributions are fundamental in statistics for describing how probabilities are assigned to the values of a random variable. This section introduces the concepts of random variables, probability distributions, and their graphical and numerical representations.

• Random Variable: A variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.

• Probability Distribution: A description that gives the probability for each value of the random variable. It can be expressed as a table, formula, or graph.

• Probability Histogram: A graph that visually depicts a probability distribution, with the vertical scale showing probabilities instead of relative frequencies.

Basic Concepts of Probability Distributions

Types of Random Variables

• Discrete Random Variable: Has a collection of values that may be finite or countably infinite (e.g., number of heads in coin tosses).

• Continuous Random Variable: Has infinitely many values, and the collection of values is not countable (e.g., heights of students).

Requirements for a Probability Distribution

Every probability distribution must satisfy the following three requirements:

1. There is a numerical (not categorical) random variable x, and its number values are associated with corresponding probabilities.

2. $\sum P(x) = 1$ where x assumes all possible values.

3. $0 \leq P(x) \leq 1$ for every individual value of the random variable x.

Example: Probability Distribution for Number of Females in Two Births

Suppose male and female births are equally likely. Let x be the number of females in two births. The probability distribution is:

x: Number of Females in Two Births	P(x)
0	0.25
1	0.50
2	0.25

• The variable x is numerical and discrete (finite values: 0, 1, 2).

• The probabilities sum to 1: $0.25 + 0.50 + 0.25 = 1$.

• Each probability is between 0 and 1.

Probability Formula

A probability distribution can also be represented by a formula. For the above example:

$P(x) = \frac{1}{2(2-x)!x!}$ (where x can be 0, 1, or 2)

Calculating for each value of x yields the same probabilities as in the table.

Non-Example: Software Piracy Table

Country	Proportion of Unlicensed Software
United States	0.17
China	0.70
India	0.58
Russia	0.64
Total	2.09

• This table does not describe a probability distribution because:

  • x is not numerical (countries are categorical).

  • The probabilities do not sum to 1.

Parameters of a Probability Distribution

Population Parameters

For probability distributions, the mean, variance, and standard deviation are parameters (since they describe a population, not a sample).

• Mean (μ): $\mu = \sum [x \cdot P(x)]$

• Variance (σ²): $\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$

• Alternative Variance Formula: $\sigma^2 = \sum [x^2 \cdot P(x)] - \mu^2$

• Standard Deviation (σ): $\sigma = \sqrt{\sum [x^2 \cdot P(x)] - \mu^2}$

Example: Calculating Mean, Variance, and Standard Deviation

x	P(x)	x·P(x)	(x-μ)²·P(x)
0	0.25	0.00	0.25
1	0.50	0.50	0.00
2	0.25	0.50	0.25
Total	1.00	1.00	0.50

• Mean: $\sum [x \cdot P(x)] = 1.0$

• Variance: $\sum [(x - \mu)^2 \cdot P(x)] = 0.5$

• Standard deviation: $\sigma = \sqrt{0.5} = 0.707 \approx 0.7$

Interpretation: In two births, the mean number of females is 1.0, the variance is 0.5, and the standard deviation is 0.7. The expected value for the number of females is also 1.0.

Discrete Probability Distributions: Significance of Values

Range Rule of Thumb for Identifying Significant Values

• Significantly low values: $\leq (\mu - 2\sigma)$

• Significantly high values: $\geq (\mu + 2\sigma)$

• Values not significant: Between $(\mu - 2\sigma)$ and $(\mu + 2\sigma)$

Note: The use of 2 in the rule of thumb is a guideline, not an absolute rule.

Significance of Results with Probabilities

• Significantly high number of successes: Probability of x or more successes is 0.05 or less.

• Significantly low number of successes: Probability of x or fewer successes is 0.05 or less.

Expected Value

Definition and Calculation

• The expected value of a discrete random variable x is denoted by E, and it is the mean value of the outcomes: $E = \mu = \sum [x \cdot P(x)]$

Example: Comparing Bets in Gambling

Suppose you have $5 to bet in a casino. Consider two bets:

• Roulette: Bet on the number 7. Probability of losing $5 is 37/38; probability of winning $175 is 1/38.

• Craps: Bet on the "pass line." Probability of losing $5 is 251/495; probability of winning $5 is 244/495.

Event	x	P(x)	x·P(x)
Lose (Roulette)	-$5	37/38	-$4.868
Win (Roulette)	$175	1/38	$4.605
Total		1	-$0.26

Event	x	P(x)	x·P(x)
Lose (Craps)	-$5	251/495	-$2.535
Win (Craps)	$5	244/495	$2.465
Total		1	-$0.07

• Interpretation: The expected value for roulette is -$0.26 per $5 bet, and for craps is -$0.07 per $5 bet. Thus, craps is the better bet in the long run, as it results in a smaller expected loss.