Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques

Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

2. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0, 1, 2, and 3.

3. Illustrate a probability distribution. What is the probability value P(x) to each value of the random variable?

P (E) = number of outcomes in the event / number of outcomes in the sample space

Probability Distribution of Discrete Random Variable

The mean of discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome X, according to its probability, P(X).

E(X) = μ = ΣXP(X)

where:

• E(X) is the mean of the outcomes x
• μ is the mean
• ΣXP(X) is the sum of each random variable value x multiplied by its own probability P(x)

Variance of Discrete Random Variable

FORMULA:

σ² = Σ[X² · P(X)] - μ²

where:

• σ² - variance of the discrete random variable
• μ - mean of the discrete random variable
• X - possible outcome
• P(X) - the probability of the outcome

Normal Probability Distribution

A normal distribution is a type of continuous probability distribution in which most data points cluster toward the middle of the range, while the rest taper off symmetrically toward either extreme. The middle of the range is also known as the mean of the distribution. The normal distribution is also known as a Gaussian distribution or probability bell curve. It is symmetric about the mean and indicates that values near the mean occur more frequently than values that are farther away from the mean.

The normal distribution, also known as the Gaussian Distribution, has the following formula:

Where:

• μ = mean
• σ=standard deviation
• π=3.14159...
• e=2.71828...

Standard Scores and the Normal Curve

This refers to a kind of transformed score that relates a raw score to the mean and standard deviation of a distribution. It is usually called a z-value.

The primary procedure in finding the area under the normal curve is to convert the normal curve of any given variable into a standardized normal curve by using the formula for standard scores.

Where:

• z = standard score
• μ = mean
• x = raw score (a given value of a particular variable)
• s = standard deviation

Probability of a Non-Standard Normal Variable

Example 1:

Given a normal distribution with μ = 200 and σ = 10, find the area above 188.

Simple Random Sampling

where:

• n = is the sample size
• N = is the population size
• e = margin of error (usually 0.01 or 0.05)

Systematic Sampling

where:

• k = sample interval
• N = population size
• n = sample size

Stratified Random Sampling

Formula: n₁ = n/k

Proportional Allocation

Formula: n₁ = (n₁/N)n

t-distribution Formula