Discrete and Continuous Random Variables, Probability Distributions, and Binomial Distributions

Random Variables

Definition and Types

A random variable is a numerical outcome resulting from a probability experiment, where the value is determined by chance. Random variables are typically denoted by capital letters such as X.

• Discrete Random Variable: Has either a finite or countably infinite number of possible values. These values can be plotted on a number line with gaps between each point.

• Continuous Random Variable: Has infinitely many possible values, which can be plotted on a line in a continuous fashion without gaps.

Examples of Random Variables

• Discrete: Number of light bulbs that burn out in a room of 10 light bulbs next year (X = 0, 1, 2, ..., 10).

• Discrete: Number of leaves on a randomly selected oak tree (X = 0, 1, 2, ...).

• Continuous: Time between calls to 911 (t > 0).

Tabular Examples

Probability distributions for simple random variables:

X	Probability
0	0.5
1	0.5

(Toss a coin once, X = number of Heads)

X	Probability
0	1/4
1	2/4
2	1/4

(Toss a coin twice, X = number of Tails)

Probability Distributions

Definition

A probability distribution of a discrete random variable provides the possible values of the variable and their corresponding probabilities. It can be represented as a table, graph, or mathematical formula.

Example Table: Movies Streamed on Netflix

x	P(x)
0	0.06
1	0.58
2	0.22
3	0.10
4	0.03
5	0.01

Properties of Probability Distributions

• For a discrete probability distribution, if P(x) denotes the probability that the random variable X takes the value x, then: $0 \\leq P(x) \\leq 1$ $\sum P(x) = 1$

Example of Invalid Probability Distribution

x	P(x)
0	0.16
1	0.18
2	0.22
3	0.10
4	0.30
5	-0.01

Additional info: Negative probabilities and probabilities summing to more than 1 indicate an invalid probability distribution.

Probability Histogram

Definition and Example

A probability histogram visually represents the probability distribution of a discrete random variable. Each bar's height corresponds to the probability of each value.

x	P(x)
0	0.06
1	0.58
2	0.22
3	0.10
4	0.03
5	0.01

Additional info: The histogram helps visualize the distribution's shape and identify the most likely outcomes.

Mean and Standard Deviation of Discrete Random Variables

Mean (Expected Value)

The mean (expected value) of a discrete random variable is given by:

$\mu_x = \sum [x \cdot P(x)]$

• x is the value of the random variable.

• P(x) is the probability of observing the value x.

Example Calculation

x	P(x)
0	0.06
1	0.58
2	0.22
3	0.10
4	0.03
5	0.01

Calculation:

$\mu_x = 0(0.06) + 1(0.58) + 2(0.22) + 3(0.10) + 4(0.03) + 5(0.01) = 1.49$

Standard Deviation

The standard deviation of a discrete random variable is given by:

$\sigma_x = \sqrt{\sum [(x - \mu_x)^2 \cdot P(x)]}$

Alternatively,

$\sigma_x = \sqrt{\sum [x^2 \cdot P(x)] - \mu_x^2}$

• x is the value of the random variable.

• \mu_x is the mean.

• P(x) is the probability of observing x.

Example Calculation

Given the same probability distribution, substitute values to find variance and standard deviation.

Binomial Distribution

Definition and Conditions

A binomial experiment is a probability experiment that satisfies the following conditions:

1. The experiment is performed a fixed number of times (n trials).

2. Each trial is independent.

3. There are two mutually exclusive outcomes: success or failure.

4. The probability of success (p) is the same for each trial.

Binomial Probability Formula

The probability of obtaining x successes in n independent trials is:

$P(x) = \binom{n}{x} p^x (1-p)^{n-x}$

• n: number of trials

• x: number of successes

• p: probability of success

• 1-p: probability of failure

Example Applications

• Probability that exactly 5 out of 20 households have three or more cars (with p = 0.35): $P(5) = \binom{20}{5} (0.35)^5 (0.65)^{15} = 0.1272$

• Probability that less than 4 have three or more cars: $P(X < 4) = P(0) + P(1) + P(2) + P(3) = 0.0444$

• Probability that at least 4 have three or more cars: $P(X \geq 4) = 1 - P(X < 4) = 1 - 0.0444 = 0.9556$

Mean and Standard Deviation of Binomial Distribution

• Mean: $\mu = n p$

• Standard Deviation: $\sigma = \sqrt{n p (1-p)}$

Example Calculation

• If 80% of a community favors a health center and 10 people are chosen, the mean number favoring the center is: $\mu = 10 \times 0.8 = 8$

• The standard deviation is: $\sigma = \sqrt{10 \times 0.8 \times 0.2} = \sqrt{1.6} \approx 1.26$

Calculator Usage

To compute binomial probabilities on a TI-84 calculator, use the binompdf function: binompdf(n, p, x) where n is the number of trials, p is the probability of success, and x is the number of successes.

Summary Table: Discrete vs. Continuous Random Variables

Type	Definition	Examples
Discrete	Finite or countable number of values	Number of heads in coin tosses, number of cars
Continuous	Infinitely many values, measurable	Time, weight, height

Additional info: These concepts form the foundation for probability theory and inferential statistics, essential for analyzing random phenomena in real-world applications.