1. Determine whether each statement is true or false. (a) (2...

1. Determine whether each statement is true or false. (a) (2 points) If Cov(X,Y) = 0, then X and Y are independent. (b) (2 points) If two events A and B are disjoint, then P(A\cap B) = P(A) + P(B). (c) (2 points) If E(XY) = E(X)E(Y), then X and Y are uncorrelated. (d) (2 points) If two events A and B are independent, then P(A\cup B) = P(A)P(B). (e) (2 points) If the joint support of random variables X and Y is not rectangular, then X and Y are dependent.

Transcript

00:01 Hello students, we have given the statement regarding the mutually exclusive events of a and b and we have to state whether they are true or false.

00:10 So, the probability of a into probability of b is greater than or equal to probability of their union.

00:16 The statement is wrong because for mutually exclusive event their intersection that is probability of intersection is equal to empty state meaning that they cannot occur simultaneously.

00:35 Therefore, probability of a union b is equal to their addition and the inequality should be reversed.

00:43 That is the correct statement is probability of a into probability of b is less than or equal to probability of a union.

00:51 For the second statement, probability of the union is equal to the probability of a plus probability of b and probability of their intersection.

01:01 So, this statement is true since for mutually exclusive event their intersection is empty state and they cannot occur together.

01:11 Therefore, the probability of their individual probability as there is no overlap.

01:16 Hence, the statement is true.

01:19 For the next problem, the odds in a favor of event b given events a, b, c are independent.

01:27 So, probability of a given b is equal to 1 by 6 and probability of a given d is equal to 4 by 9.

01:36 Since a is a proper subset of b, probability of a is a part of probability of d.

01:42 Therefore, probability of d equal to probability of a plus probability of a dash intersection d which is equal to probability of a plus probability of d given a which after before that we need to find the probability of c and probability of d.

02:01 Since we have provided that probability of c is equal to probability of d...

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