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Let X be a random variable with probability density function f(x) = { λe^(−λx) 0 ≤...

Let X be a random variable with probability density function f(x) = { λe^(−λx) 0 ≤ x < ∞

0 otherwise } for some λ > 0.

a. Compute the cumulative distribution function F(x), where F(x) = Prob(X < x) viewed as a function of x.

b. The α-percentile of a random variable is the number mα such that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the random variable X. The value of mα will be a function of α and λ.

c. Let X be the random variable. For any a > 0 and b > 0, compute Prob(X > a + b|X > a)

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