April 11, 2009 - Lecture Notes

Certain applications, such as those used for weather forecasting and space research, require the
collection and analysis of large amounts of data.

For such applications, data is often collected using probability experiments and the outcome of these experiments is organized to form probability distributions. The shape, central tendency, and variability of probability distributions help analyzers
find patterns in the data set and make predictions and decisions.

You will be introduced to discrete probability distributions—a specific type of probability
distribution. Continuous probability distributions will be covered later in the course

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A. Probability Distributions

A random variable, x, represents a numerical value associated with each outcome of a probability experiment.

A discrete random variable is a random variable with countable possible outcomes that can be listed. A continuous random variable is a random variable with an uncountable number of possible outcomes represented by an interval on the number line.

B. Binomial Distributions
Binomial experiments produce only two outcomes per trial, often called Success S and Failure F.
Examples include the possible outcomes when flipping a coin or answering a question with two answer options.

A binomial experiment is a discrete probability experiment that must satisfy the four conditions given below:

Condition 1: The experiment is repeated for a fixed number of trials, n. For example, a coin is
flipped 10 times. Each trial is independent of the other trials.

Condition 2: There are only two possible outcomes of interest for each trial. One of these outcomes is classified as a success (S) and the other as a failure (F). For example, flipping a coin has two possible outcomes—heads or tails. You may consider the occurrence of heads as a success and tails as a failure.

Condition 3: The probability of a success P(S) and the probability of a failure P(F) is the same for
each trial. For example, the probability of getting a six when tossing a fair dice is 1 6
and the probability of not getting a six is 5 6 and these probabilities are the same regardless of how many times the dice is thrown.

Condition 4: The random variable x counts the number of successful trials in the total number of
trials, n: x = 0, 1, 2, 3, …, n. If six is the result two times in 10 flips, x = 2.