MATHS 2201 Engineering Mathematics I Tutorial Exercise 2Week of 12 March 2012

1. Suppose the diameters of pistons produced by a certain manufacturer are normally distributedwith mean 120mm and standard deviation 0.2mm. The diameters of the corresponding cylindersare normally distributed with mean 120.5mm and standard deviation 0.1mm. Find the probabilitythat a randomly chosen piston will be too large to fit in a randomly chosen cylinder.

2. Let the time to failure of an electronic component, T , be an exponential random variable with rateλ. Given that the component has not failed in r minutes, what is the probability that it will notfail in the next s minutes? Explain why this is called the ‘memoryless’ property of the exponentialdistribution.

3. Suppose X1 and X2 are independent random variables with E(X1) = µ1, var(X1) = σ21 , E(X2) =

µ2 and var(X2) = σ22 , and let

D = X2 −X1.

(a) Find E(D) and var(D).

(b) Show that cov(X1, D) = −σ21 .

(c) Find the correlation between X1 and D.

(d) In a sports engineering experiment, two identical footballs were used, one filled with ordinaryair and the other filled with helium. A footballer then kicked the air filled football followed bythe helium filled football from 39 different positions on the field and the distance of each kick(in yards) was recorded. Shown below is a scatter plot of the difference between the secondand first kicks against the distance of first kick.

Note that a positive difference means that the second kick (helium filled) was longer and anegative difference means that the first kick (air filled) was longer.

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

15 20 25 30 35

−15

−5

05

10

Distances of football kicks

First Kick

Diff

eren

ce

One explanation for the pattern in the scatter plot is that:

• If the first kick was short, then the footballer would try harder with the second kick.

• If the first kick was long, the footballer would be fatigued and less able to kick as far thesecond time.

Explain whether this would be a reasonable conclusion.

1

4. The total cost of repair for an electrical appliance is determined by the price of the parts and thelabour. At a certain company, it is known from previous records that for a randomly chosen repairjob:

• The cost of the parts X has mean µX = $185 and standard deviation σX = $62;

• The amount of labour T has mean µT = 1.25 hours and standard deviation σT = 0.3 hours.

• The correlation coefficient is ρXT = 0.31.

• The cost of labour is $85 per hour.

(a) Give an expression for the total cost of the repair, R, in terms of X and T .

(b) Find µR and σR.

5. An MP3 player uses a single AAA battery and the lifetime X for a certain brand of alkaline batteryhas E(X) = 9.5 hours and var(X) = 0.752. Find the probability that a packet of 25 batteries willbe sufficient to power the player for at least 240 hours in total. State any assumptions you havemade in calculating your answer.

6. Suppose the actual weights in kg for bags of fertiliser produced by a certain manufacturer havethe uniform distribution U(39.5, 41.0). Suppose also that pallet is to be loaded with 25 randomlychosen bags of fertiliser let W be the total weight of the bags of fertiliser (excluding the weight ofthe pallet).

(a) If X has the uniform distribution, X ∼ U(39.5, 41.0) find µ = E(X) and σ2 = var(X).

(b) Find µW and σW .

(c) Find an approximate value for the probability that the total weight is between 1005 and1010kg. Clearly state any assumptions you have made in your caculations.

7. The lifetime of a certain spacecraft component is normally distributed with mean µ = 3000 hoursand standard deviation σ = 800 hours. A failed component can be replaced immediately by a newone during a mission. What is the probability that one spare will suffice for a mission of 3000 hoursis the component lifetimes are assumed independent.

? ? ? ? ? ? ?

2