Problem 6 (p.182#10). Let A and B be independent events with indicator random variables $I_A$ and $I_B$. a) Describe the distribution of the random variable $(I_A + I_B)^2$ in terms of $P(A)$ and $P(B)$. b) What is $E[(I_A + I_B)^2]$?
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This means: $I_A = 1$ if event A occurs, and $I_A = 0$ if event A does not occur. $I_B = 1$ if event B occurs, and $I_B = 0$ if event B does not occur. The probabilities are $P(I_A = 1) = P(A)$ and $P(I_A = 0) = 1 - P(A)$. The probabilities are $P(I_B = 1) = P(B)$ Show more…
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Let X be a non-negative random variable with finite, non-zero expectation μ. Which one of the following statements is correct for every t > 0? (a) P(X ≥ tμ) ≥ 1/(tμ) (b) P(X ≥ tμ) ≤ 1/t (c) P(X ≤ tμ) ≤ 1/(tμ) (d) P(X ≤ tμ) ≥ 1/t 10. Consider two events A and B such that P(A) = 1/4, P(B|A) = 1/2 and P(A|B) = 1/4. Define random variable X as X = 1 if event A occurs and 0 if event A does not occur. Define random variable Y as Y = 1 if event B occurs and 0 if event B does not occur. Then consider the following statements : 1. P(X^2 + Y^2 = 1) = 1/4 2. P(XY = X^2Y^2) = 1 Which of the above statements is/are correct ? (a) 1 only (b) 2 only (c) Both 1 and 2 (d) Neither 1 nor 2
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