Linear combinations of independent normal variables Problem set B

X and Y are independent, \(X\sim N(25,\,7)\) and \(Y\sim N(11,\,2)\). State the distribution of each combination.

1. (a) \(W=2X+3Y\)
2. (b) \(W=4X-3Y-2\)

X and Y are independent, \(X\sim N(35,\,10)\) and \(Y\sim N(14,\,3)\). Let \(W=2X-3Y+4\). Find \(P(W>40)\).

X has mean 60 and standard deviation 5. Y has mean 45 and standard deviation 2. X and Y are independent and both are normal. State the distribution of each.

1. (a) \(X+Y\)
2. (b) \(X-Y\)

A student gets ready for school and then walks there. The getting-ready time, X minutes, has distribution \(N(25,\,16)\) and the walking time, Y minutes, has distribution \(N(18,\,9)\). The two times are independent. Let T = X + Y. State the distribution of T and find the probability that the student needs less than 40 minutes in total.

A photographer charges \$60 for each hour of shooting and \$35 for each hour of editing. The shooting time for a job, X hours, has distribution \(N(3.5,\,0.8)\). The editing time, Y hours, has distribution \(N(5,\,2.25)\). The two times are independent. The cost of a job is C = 60X + 35Y dollars. State the distribution of C and find the probability that a job costs more than \$420.

Water flows into a tank through one pipe and out through another over one hour. The inflow, X litres, has distribution \(N(210,\,64)\) and the outflow, Y litres, has distribution \(N(185,\,49)\). The two flows are independent. Let D = X − Y be the net gain in the tank. State the distribution of D and find the probability that the tank gains more than 35 litres.

X, Y and Z are independent, \(X\sim N(15,\,5)\), \(Y\sim N(9,\,2)\) and \(Z\sim N(12,\,3)\). Let \(W=X+3Y-2Z-4\). Find \(P(W<22)\).

X and Y are independent, \(X\sim N(50,\,16)\) and \(Y\sim N(30,\,4)\).

1. (a) State the distributions of \(X+Y\) and \(X-Y\).
2. (b) Find \(P(X+Y>85)\).
3. (c) Find \(P(X-Y>25)\). Explain why your answer is the same as in part (b).