Let Z be the standard normal distribution. Suppose A={Z > 0} and B= {-1>Z1}. Explain why A and B are independent.
-
To explain why A and B are independent, we just need to show that
P(A and B) = P(A)P(B).
P(A) = P(Z > 0) = 1/2, by symmetry of the standard normal distribution about z-score zero, combined with the fact that P(Z = 0) is zero because Z is a continuous random variable.
P(B) = P(-1 < Z < 1) = P(-1 < Z < 0) + P(Z = 0) + P(0 < Z < 1) = 2P(0 < Z < 1),
since P(-1 < Z < 0) = P(0 < Z < 1) by symmetry of the standard normal distribution about z-score zero, and P(Z = 0) is zero because Z is a continuous random variable.
So P(A)P(B) = (1/2)[2P(0 < Z < 1)] = P(0 < Z < 1).
P(A and B) = P(Z > 0 and -1 < Z < 1) = P(0 < Z < 1) = P(A)P(B).
We conclude that A and B are independent.
Lord bless you today!
P(A and B) = P(A)P(B).
P(A) = P(Z > 0) = 1/2, by symmetry of the standard normal distribution about z-score zero, combined with the fact that P(Z = 0) is zero because Z is a continuous random variable.
P(B) = P(-1 < Z < 1) = P(-1 < Z < 0) + P(Z = 0) + P(0 < Z < 1) = 2P(0 < Z < 1),
since P(-1 < Z < 0) = P(0 < Z < 1) by symmetry of the standard normal distribution about z-score zero, and P(Z = 0) is zero because Z is a continuous random variable.
So P(A)P(B) = (1/2)[2P(0 < Z < 1)] = P(0 < Z < 1).
P(A and B) = P(Z > 0 and -1 < Z < 1) = P(0 < Z < 1) = P(A)P(B).
We conclude that A and B are independent.
Lord bless you today!