Joint Probability Distribution(123).pptx

joint probability distribution

1. Joint Probability Distribution Prepared by: Antipolo, Laurece Xedrex Calumpiano , Rica Cenal , James Lourenz S. Enero , Julius David Villamor, Aaliyah Marie E. GROUP 5

2. Joint Probability Distribution Given two random variable that are defined on the same probability space, the  joint probability distribution  is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distribution, i.e. the distributions of each of the individual random variables and the conditional probability distribution, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s). Many samples observations (black) are shown from a joint probability distribution. The marginal densities are shown as well (in blue and in red).

3. Formula for Joint Probability Notation to represent the joint probability can take a few different forms. The following formula represents the joint probability of events with intersection. P (A⋂B) where, A, B= Two events P(A and B),P(AB)=The joint probability of A and B The symbol “∩” in a joint probability is called an intersection. The probability of event A and event B happening is the same thing as the point where A and B intersect. Hence, the joint probability is also called the intersection of two or more events. We can represent this relation using a Venn diagram as shown below.

4. EXAMPLE: Find the probability that the number three will occur twice when two dice are rolled at the same time. Solution: Number of possible outcomes when a die is rolled = 6 i.e. {1, 2, 3, 4, 5, 6} Let A be the event of occurring 3 on first die and B be the event of occurring 3 on the second die. Both the dice have six possible outcomes, the probability of a three occurring on each die is 1/6. P(A) =1/6 P(B )=1/6 P(A,B) = 1/6 x 1/6 = 1/36

5. Joint Probability Table A joint probability distribution represents a probability distribution for two or more random variables. Instead of events being labelled A and B, the condition is to use X and Y as given below. f( x,y ) = P(X = x, Y = y) The main purpose of this is to look for a relationship between two variables. For example, the below table shows some probabilities for events X and Y happening at the same time: This table can be used to find the probabilities of events.

6. EXAMPLE: Find the probability of X = 3 and Y = 3. Solution:  From this table, identify the probability under X = 3 and Y= 3. That is ⅙.

7. Joint Probability Mass Function Remember that for a discrete random variable X, we define the PMF as P(x)=P(X=x). Now, if we have two random variable X and Y, and we would like to study them jointly, we define the joint probability mass function as follows: The joint probability mass function of two discrete random variables X and Y is defined as: P XY ( x,y )= P(X=x, Y=y). = P((X=x) Ո (Y=y))

8. Example:

9. Joint Probability Density Function   Consider the joint pdf of two variables

10. Marginal Probability Distribution Once a joint probability mass function for (X,Y) has been constructed, one finds probabilities for one of the two variables. Consider a discrete random vector, that is, a vector whose entries are discrete random variables. When one of these entries is taken in isolation, its distribution can be characterized in terms of its probability mass function. This is called marginal probability mass function, in order to distinguish it from the joint probability mass function, which is instead used to characterize the joint distribution of all the entries of the random vector considered together.

11. Conditional Probability Distribution It is the probability distribution of a random variable, calculated according to the rules of conditional probability after observing the realization of another random variable. In other words, a conditional probability distribution describes the probability that a randomly selected person from a sub-population has a given characteristic of interest.

12. More than Two Random Variables

13. EXAMPLE:

14. Linear Function of Random Variable If X is a random variable and Y is a linear function of the random variable X, where:

15. General Function of Random Variable

16. Thank you!!!