(i) P {X < a} = P {X > a} and (ü) P {X > b} = 0.05

If the probability density of X is given by f(x) =2x−3 for x > 10 elsewherecheck whether its mean and its variance exist.

The lifetime X in hours of an electronic tube is a random variable having a probability density function given by f(r) = re, I (a) Compute the expected lifetime of such a tube. (b) Compute the variance of the lifetime of such a tube. (c) Find the median lifetime m of such a tube that satisfies P(X > m) = 0.5.

The shelf life (in hours) of a certain perishable pack-aged food is a random variable whose probability density function is given byf(x) =⎧⎪⎪⎨⎪⎪⎩20,000(x + 100)3 for x > 00 elsewhere Find the probabilities that one of these packages will havea shelf life of(a) at least 200 hours;(b) at most 100 hours;(c) anywhere from 80 to 120 hours.

A continuous random variable X has probability distribution function f(x) , wheref(x) = k(1 − x2) for − 1 < x < 1.Find the variance of the random variable X.

4. The monthly revenue (in millions) of a local gaming company is represented by a continuous random variable X having the probability density f(x) = (1 - (x-1)²), 0 < x < 2, 0, elsewhere Find the mean and variance of X.

2. The time (in minutes) required for technicians to perform a specific task is believed to follow an exponential distribution with probability density function 0.2e-0.2t if t>0 { f(t) = 0ift<0 (a) What is the mean time required by technicians to perform the task and its variance? (b) Calculate the probability that a technician can complete the task in less than 7 mi Post