Linear combinations of independent normal variables Problem set A

X and Y are independent, \(X\sim N(18,\,5)\) and \(Y\sim N(6,\,3)\). State the distribution of each combination.

1. (a) \(W=X+4Y\)
2. (b) \(W=3X-2Y+7\)

X and Y are independent, \(X\sim N(22,\,6)\) and \(Y\sim N(9,\,4)\). Let \(W=3X-2Y+5\). Find \(P(W>50)\).

X has mean 50 and standard deviation 4. Y has mean 30 and standard deviation 3. X and Y are independent and both are normal. State the distribution of each.

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

A delivery driver loads the van and then drives to the customer. The loading time, X minutes, has distribution \(N(18,\,9)\) and the journey time, Y minutes, has distribution \(N(41,\,36)\). The two times are independent. Let T = X + Y. State the distribution of T and find the probability that the whole delivery takes less than 65 minutes.

A plumber charges \$18 for each hour of labour and \$40 for each hour of equipment hire. The labour time on a job, X hours, has distribution \(N(5.5,\,1.2)\). The equipment time, Y hours, has distribution \(N(2.5,\,0.5)\). The two times are independent. The cost of a job is C = 18X + 40Y dollars. State the distribution of C and find the probability that a job costs more than \$250.

A cyclist climbs a hill and then rides back down. The time to climb, X minutes, has distribution \(N(14.5,\,2.25)\) and the time to descend, Y minutes, has distribution \(N(6.2,\,0.64)\). The two times are independent. Let D = X − Y. State the distribution of D and find the probability that the climb takes more than 8 minutes longer than the descent.

X, Y and Z are independent, \(X\sim N(10,\,4)\), \(Y\sim N(6,\,3)\) and \(Z\sim N(8,\,2)\). Let \(W=X+2Y-Z+1\). Find \(P(W<18)\).

X and Y are independent, \(X\sim N(40,\,9)\) and \(Y\sim N(25,\,16)\).

1. (a) State the distributions of \(X+Y\) and \(X-Y\).
2. (b) Find \(P(X+Y>70)\).
3. (c) Find \(P(X-Y>20)\). Explain why your answer is the same as in part (b).