Way 2: Defining a new random variable X as the amount earned by the private debt collector per week. Then we can define X = + Y since X can be expressed as a linear function of Y. The probability distribution of X can be found as follows (start from the smallest realization of X) :

For way 2, E (X) = $175

Step-by-step explanation:

Y is the number of loans collected in a week

given that $2000 is spent each week and charges $1500 per loan collected as service,

Therefore; the total amount earned (or lost) each week by the collector

X = - 2000 + 1500*Y

In finding the probability distribution of X, the possible values of Y are 0,1,2 and 3

Thus, the possible values of X are:

when X = 0, - 2000 + 1500*0 = - 2000

X = 1, - 2000 + 1500*1 = - 500

X = 2, - 2000 + 1500*2 = 1000

X = 3, - 2000 + 1500*3 = 2500.

Now, P (X = - 2000) = P (-2000 + 1500*Y = - 2000) = P (Y=0) = 0.15

P (X = - 500) = P (-2000 + 1500*Y = - 500) = P (Y=1) = 0.40

P (X = 1000) = P (-2000 + 1500*Y = 1000) = P (Y=2) = 0.30

P (X = 2500) = P (-2000 + 1500*Y = 2500) = P (Y=3) = 0.15

Thus, the probability distribution of X is given by:

X - 2000 - 500 1000 2500

p (x) 0.15 0.40 0.30 0.15

see in the workings below the expected value of X

E (X) = Σₓx * P (X = x)

= (-2000) * P (X=-2000) + (-500) * P (X=-500) + 1000*P (X=1000) + 2500*P (X=2500)

= (-2000) * 0.15 + (-500) * 0.40 + 1000 * 0.30 + 2500*0.15

= - 300 - 200 + 300 + 375

= 175