Solve the following

A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by f(x) = 2(1 + x)/27. Find (a) P(X 0, y>0, elsewhere, find P(0 0, y>0, elsewhere, Find -(sty). r>0, y>0, f(x,y) = 10, elsewhere,From a box containing 4 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls. A typical roulette wheel used in a casino has 38 slots that are numbered 1, 2, 3, . . ., 36, 0, 00, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of 1/38, and we are interested in the number of the slot into which the ball falls. (a) Define the sample space S. (b) Let A = {0, 00). Give the value of P (A). (c) Let B = {14, 15, 17, 18). Give the value of P (B). (d) Let D = {x : x is odd). Give the value of P (D)