Exercise III: Improper integrals in probability Consider a function f dened on [0, +90}: and assume that n x} 2 0 for all r; n the integral 15\": Hr] d1? is convergent and oo D f:)d.7:=l. This is called a enmity function in probability. Its interpretation is that the probability that a random result is between a and b 2 a is j: r) d:. It does not matter if \"between" includes a or b, and i: could be innite. Given such a density function, the quantity E = L+xrf(r]dr is called the emaciaticn: it describes the average of the random rwults when we repeat the experiment many times. Now, we can compute M = ['3' end: D and then the torsional? V = M E2. The variance describes how spread apart the values are. 1. We rst take HI} = ore-aI for or > D. This is known as the exponential distribution. Check that it is a density function, that is that ac {(1) :11 = 1. D "hen compute its expectation and variance. 2. The exponential distribution is often used to model a waiting; time. Say that you take the bus to school every morning, and notice that you wait on average 10 minutes. What is the probability that you have to wait less than 5 minutes? More than 10? More than 21]? Give all results in exact form, then as a percentage rounded to the nearest integer. 3. Recall that we saw in HW' 4 that +00 -12 _l .. j; e dtz. We now take 2 2 KEY, fir) =1]. 023, where a > D is a constant. Check that this is a density function, and compute its mean and variance. Hint: usub and IBP