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  -2000-50010002500

p(x)   0.150.400.300.15

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

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the forst and 3rd

step-by-step explanation: