Let F(x) be the cdf of the continuous-type random variable X, and assume that F(x) = 0

Chapter 3, Problem 3.2-4

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QUESTION:

Let F(x) be the cdf of the continuous-type random variable X, and assume that F(x) = 0 for x 0 and 0 < F(x) < 1 for 0 < x. Prove that if P(X > x + y | X > x) = P(X > y), then F(x) = 1 ex, 0 < x. Hint: Show that g(x) = 1 F(x) satisfies the functional equation g(x + y) = g(x)g(y), which implies that g(x) = acx.

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QUESTION:

Let F(x) be the cdf of the continuous-type random variable X, and assume that F(x) = 0 for x 0 and 0 < F(x) < 1 for 0 < x. Prove that if P(X > x + y | X > x) = P(X > y), then F(x) = 1 ex, 0 < x. Hint: Show that g(x) = 1 F(x) satisfies the functional equation g(x + y) = g(x)g(y), which implies that g(x) = acx.

ANSWER:

Step 1 of 4

Given that,

Let  be the cdf of the continuous-type random variable X, and assume that  for  and  for .

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