Question

Exercise 2.26. Suppose events A,B,C,D are mutually independent. Show that events AB and CD are independent. Justify each step from the definition of mutual independence.

Answer 1

To show AB and CD are independent events, we used the fact that A, B, C, and D are mutually independent. The calculations show that P(AB AND CD) equals P(AB)P(CD), satisfying the condition for independence of events AB and CD.

Step-by-step explanation:

To demonstrate that events AB and CD are independent, we need to use the definition of mutual independence. By this definition, being mutually independent, events A, B, C, and D satisfy the condition that for any two distinct events, say A and B, P(A AND B) = P(A)P(B). Similarly, this extends to any three events and all four events together, giving us P(A AND B AND C) = P(A)P(B)P(C), and P(A AND B AND C AND D) = P(A)P(B)P(C)P(D).

Sine AB and CD are composed of mutually independent events, we can deduce the following:

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Now, to show that AB and CD are independent, we need to verify if P(AB AND CD) = P(AB)P(CD).

Because A, B, C, and D are mutually independent, we can expand P(AB AND CD) as:

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And since we already know the individual probabilities of AB and CD as shown above, we then have:

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Observing that both P(AB AND CD) and P(AB)P(CD) result in the same product P(A)P(B)P(C)P(D), we conclude that AB and CD are indeed independent events.