What is the variance of the difference between two indicator random variables, IA and IB, in terms of the probabilities of events A and B, P(A) and P(B), and the probability of both events A and B occurring, P(A∩B)?
To calculate the variance of the difference between two indicator random variables, IA and IB, we need to understand the properties and definitions of indicator random variables.
Indicator random variables take on the value of 1 if a certain event occurs, and 0 otherwise. In this case, IA takes the value of 1 if event A occurs, and 0 otherwise. Similarly, IB takes the value of 1 if event B occurs, and 0 otherwise.
The difference between IA and IB is defined as (IA - IB).
First, let's calculate the expected value of (IA - IB):
The expected value (E) of a random variable is the sum of the products of each possible value of the random variable and its corresponding probability. Since IA and IB are indicator random variables, both can only take values of 0 or 1.
E(IA - IB) = (0 * P(IA = 0, IB = 1)) + (1 * P(IA = 1, IB = 0)) + (0 * P(IA = 1, IB = 1)) + (1 * P(IA = 0, IB = 0))
Since IA and IB are independent random variables, the joint probability P(IA = x, IB = y) is equal to the product of the individual probabilities:
E(IA - IB) = (0 * P(IA = 0) * P(IB = 1)) + (1 * P(IA = 1) * P(IB = 0)) + (0 * P(IA = 1) * P(IB = 1)) + (1 * P(IA = 0) * P(IB = 0))
Simplifying further:
E(IA - IB) = P(IA = 1) * P(IB = 0) - P(IA = 0) * P(IB = 1)
Next, we can calculate the variance of (IA - IB):
The variance (Var) of a random variable is a measure of how spread out its values are. It is defined as the expected value of the squared difference between each value of the random variable and its expected value.
Var(IA - IB) = E((IA - IB - E(IA - IB))^2)
Expanding the square and simplifying, we get:
Var(IA - IB) = E((IA - IB)^2 - 2(IA - IB)E(IA - IB) + (E(IA - IB))^2)
Since IA and IB take on values of 0 or 1, (IA - IB)^2 simplifies to IA^2 - 2IAIB + IB^2. Since IA and IB are indicator random variables, their squares are equal to themselves:
Var(IA - IB) = E(IA - 2IAIB + IB - 2IAIB + 2(IB)^2) - (E(IA - IB))^2
Since IA and IB can only take on values of 0 or 1, their squares are equal to themselves:
Var(IA - IB) = E(IA - 2IAIB + IB - 2IAIB + 2IB) - (E(IA - IB))^2
Expanding further:
Var(IA - IB) = E(IA) - 2E(IAIB) + E(IB) - 2E(IAIB) + 2E(IB) - (E(IA - IB))^2
Evaluating the corresponding probabilities gives:
Var(IA - IB) = P(A) - 2P(A∩B) + P(B) - 2(P(A) - P(A∩B)) + 2P(B) - (P(A) - P(A∩B) - P(B))^2
Simplifying:
Var(IA - IB) = P(A) + P(B) - 2P(A∩B) - (P(A) - P(A∩B) - P(B))^2
Thus, the variance of the difference between two indicator random variables IA and IB, in terms of the probabilities of events A and B, P(A) and P(B), and the probability of both events A and B occurring, P(A∩B), is given by P(A) + P(B) - 2P(A∩B) - (P(A) - P(A∩B) - P(B))^2.