If a random variable has the probability density f(x)= [2e-2x for x>0 for x≤0 0 find the probabilities that it will take on a value a) between 1 and 3 b) greater than 0.5

Suppose x is a random variable best described by a uniform probability that ranges from 0 to 4. Compute the following: (a) the probability density function f(x) = ☐ for 0 ≤ x ≤ 4 (b) the mean µ = (c) the standard deviation σ = (d) P(μ-o≤ x ≤ μ + 0) = (e) P(x > 1.76) =

Suppose that a random variable X has the following probability density function. SC (64-x²) 0 ≤ x ≤ 8 otherwise f(x) = {C Find the expected value of X. (You will need to find the value of the constant C so that f is a pdf.)

The probability that none of the three bulbs in a set of traffic lights will have to be replaced during the first 1200 hours of operation if the lifetime X of a bulb (in thousands of hours) is a random variable with probability density function f(x) = 6[0.25-(x-1.5)2] when 1 ≤ x ≤ 2 and f(x) = 0 otherwise is