Question

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):

Answer 1

For way 2, E(X) = $175

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