Independence of Events
Independence in probability theory refers to the scenario where the occurrence of one event does not influence the occurrence of another. When two events, A and B, are independent, the probability that both events occur is the product of their individual probabilities:
P(A∩B) = P(A)⋅P(B)
Example: Consider rolling a die and flipping a coin. The result of rolling a die (say, getting a 4) does not impact the result of flipping a coin (getting heads). Here:
- A: Rolling a 4 on a die.
- B: Getting heads on a coin flip.
If P(A)=1/6 and P(B)=1/2, then:
P(A∩B)=1/6⋅1/2=1/12.
Identically Distributed
Identically Distributed refers to two or more random variables having the same probability distribution. If X and Y are two random variables with the same distribution, their probability density functions (PDFs) or probability mass functions (PMFs) are identical:
P(X=x) = P(Y=y)
Example: Consider the heights of students in two different classrooms, where both classrooms have students of the same age group and demographic characteristics. Let:
- X: Heights of students in classroom A.
- Y: Heights of students in classroom B.
If the distribution of heights in both classrooms is similar, we say that X and Y are identically distributed.
Imagine two bell curves (normal distributions) with the same mean and standard deviation. These curves represent the distribution of X and Y, showing that they are identically distributed.
Problem: Suppose you conduct an experiment where you roll two fair six-sided dice. Let A be the event of rolling a 3 on the first die, and B be the event of rolling a 6 on the second die. Also, let X and Y be the sum of the numbers on each die over several trials.
- Are A and B independent events?
- Are X and Y identically distributed?
Solution:
1. Independence:
P(A) = 1/6
P(B) = 1/6
Since the outcome of rolling the first die does not affect the outcome of rolling the second die:
P(A∩B) = P(A)⋅P(B) = 1/6⋅1/6 = 1/36. Thus, A and B are independent events.
2. Identically Distributed:
The sum of the numbers on each die, X and Y, follows the same distribution. Both are uniformly distributed over {1, 2, 3, 4, 5, 6}.Therefore, X and Y are identically distributed random variables.