Problem 1: A random variable X taking values between A = 1 and B= 2 has the probabili density function f(=) = ,15z52. (a) Check that f is indeed a probability density function. (b) Compute the cumulative distribution function F. (c) Use your result from (b) to compute Pr(1 < X < }).

Question 1.2 Consider the function  f (x) = (1/24(x^2 +1)      1 < or =  x < or = 4)          = (0      otherwise)   Calculate P (x = 3) Calculate P (2 < or =  x < or = 3)         Question 1.3 Consider the function  f (x) = (k - x/4      1 < or =  x < or = 3)          = (0      otherwise)   which is being used as a probability density function for a continuous random variable x?   a. Find the value of K b. Find P (x < or = 2.5)

A machine has a useful life of 4 to 9 years, and its life (in years) has a probability density function defined by f(x) = 1 1 Find the probability that the useful life of such a machine selected 15 at random will be longer than 7 years?