Statistics Tutorial: Probability Distributions
What Does Probability Distribution Mean?
A statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. This range will be between the minimum and maximum statistically possible values, but where the possible value is likely to be plotted on the probability distribution depends on a number of factors, including the distributions mean, standard deviation, skewness and kurtosis.o understand probability distributions, it is the important to understand variables. random variables, and some notation.
- A variable is a symbol (A, B, x, y, etc.) that can take on any of a specified set of values.
- When the value of a variable is the outcome of a statistical experiment, that variable is arandom variable.
Generally, statisticians use a capital letter to represent a random variable and a lower-case letter, to represent one of its values. For example,
- X represents the random variable X.
- P(X) represents the probability of X.
- P(X = x) refers to the probability that the random variable X is equal to a particular value, denoted by x. As an example, P(X = 1) refers to the probability that the random variable X is equal to 1.
Probability Distributions
An example will make clear the relationship between random variables and probability distributions. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of Heads that result from this experiment. The variable X can take on the values 0, 1, or 2. In this example, X is a random variable; because its value is determined by the outcome of a statistical experiment.
A probability distribution is a table or an equation that links each outcome of a statistical experiment with its probability of occurence. Consider the coin flip experiment described above. The table below, which associates each outcome with its probability, is an example of a probability distribution.
| Number of heads | Probability |
|---|
| 0 | 0.25 |
| 1 | 0.50 |
| 2 | 0.25 |
The above table represents the probability distribution of the random variable X.
Cumulative Probability Distributions
A cumulative probability refers to the probability that the value of a random variable falls within a specified range.
Let us return to the coin flip experiment. If we flip a coin two times, we might ask: What is the probability that the coin flips would result in one or fewer heads? The answer would be a cumulative probability. It would be the probability that the coin flip experiment results in zero heads plus the probability that the experiment results in one head.
P(X < 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75
Like a probability distribution, a cumulative probability distribution can be represented by a table or an equation. In the table below, the cumulative probability refers to the probability than the random variable X is less than or equal to x.
| Number of heads: x | Probability: P(X = x) | Cumulative Probability: P(X < x) |
|---|
| 0 | 0.25 | 0.25 |
| 1 | 0.50 | 0.75 |
| 2 | 0.25 | 1.00 |
Uniform Probability Distribution
The simplest probability distribution occurs when all of the values of a random variable occur with equal probability. This probability distribution is called the uniform distribution.
Uniform Distribution. Suppose the random variable X can assume k different values. Suppose also that the P(X = x
k) is constant. Then,
P(X = xk) = 1/k
Example 1
Suppose a die is tossed. What is the probability that the die will land on 6 ?
Solution: When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is a random variable (X), and each outcome is equally likely to occur. Thus, we have a uniform distribution. Therefore, the P(X = 6) = 1/6.
Example 2
Suppose we repeat the dice tossing experiment described in Example 1. This time, we ask what is the probability that the die will land on a number that is smaller than 5 ?
Solution: When a die is tossed, there are 6 possible outcomes represented by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is equally likely to occur. Thus, we have a uniform distribution.
This problem involves a cumulative probability. The probability that the die will land on a number smaller than 5 is equal to:
P( X < 5 ) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1/6 + 1/6 + 1/6 + 1/6 = 2/3
The distribution not only tells the company that the number of orders have ranged from 41 to 48 orders per week, but also that most weeks have between 43 and 46 orders. Another way to illustrate this probability distribution is in the form of a graph or visual depiction. A graph for the weekly order volumes is shown here.
This graph shows the probability of each number of weekly orders. The height of each bar displays the probability associated with that number of orders per week. You may notice that in the graph, it is much easier to see the cluster of orders in the range of 43 to 46 orders per week than it was in the table of values.
