Probability Distribution of a Random Variable
Now the question comes, how to describe the behavior of a random variable?
Suppose that our Random Variable only takes finite values, like x1, x2, x3 …. and xn. ie. the range of X is the set of n values is {x1, x2, x3 …. and xn}.
Thus, the behavior of X is completely described by giving probabilities for all the values of the random variable X
Event | Probability |
---|---|
x1 | P(X = x1) |
x2 | P(X = x2) |
x3 | P(X = x3) |
The Probability Function of a discrete random variable X is the function p(x) satisfying
P(x) = P(X = x)
Let’s look at an example:
Example: We draw two cards successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of finding aces.
Answer:
Let’s define a random variable “X”, which means number of aces. So since we are only drawing two cards from the deck, X can only take three values: 0, 1 and 2. We also know that, we are drawing cards with replacement which means that the two draws can be considered an independent experiments.
P(X = 0) = P(both cards are non-aces)
= P(non-ace) x P(non-ace)
= [Tex]\frac{48}{52} \times \frac{48}{52} = \frac {144}{169} [/Tex]
P(X = 1) = P(one of the cards in ace)
= P(non-ace and then ace) + P(ace and then non-ace)
= P(non-ace) x P(ace) + P(ace) x P(non-ace)
= [Tex]\frac{48}{52} \times \frac{4}{52} + \frac{4}{52} \times \frac{48}{52} = \frac{24}{169} [/Tex]
P(X = 2) = P(Both the cards are aces)
= P(ace) x P(ace)
= [Tex]\frac{4}{52} \times \frac{4}{52} = \frac{1}{169} [/Tex]
Now we have probabilities for each value of random variable. Since it is discrete, we can make a table to represent this distribution. The table is given below.
X | 0 | 1 | 2 |
---|---|---|---|
P(X=x) | [Tex]\frac{144}{169} [/Tex] | [Tex]\frac{24}{169} [/Tex] | [Tex]\frac{1}{169} [/Tex] |
It should be noted here that each value of P(X=x) is greater than zero and the sum of all P(X=x) is equal to 1.
Probability Distribution – Function, Formula, Table
A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.
Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.
A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.
In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.
Table of Content
- What is Probability Distribution?
- Probability Distribution Definition
- Random Variables
- Random Variable Definition
- Types of Random Variables in Probability Distribution
- Probability Distribution of a Random Variable
- Probability Distribution Formulas
- Expectation (Mean) and Variance of a Random Variable
- Expectation
- Variance
- Different Types of Probability Distributions
- Discrete Probability Distributions
- Bernoulli Trials and Binomial Distributions
- Binomial Distribution
- Cumulative Probability Distribution
- Probability Distribution Function
- Probability Distribution Table
- Prior Probability
- Posterior Probability
- Solved Questions on Probability Distribution