Question

A consultant has three sources of income—from teaching short​ courses, from selling computer​ software, and from advising on projects. His expected annual incomes from these sources are​$40,000​, ​$25,000​, and ​$18,000​, and the respective standard deviations are ​$4,000​, $5,000​, and $6,000.

Assuming​ independence, find the mean and standard deviation of his total annual income.

μW=​$

​(Round to the nearest dollar as​ needed.)

σW=​$

​(Round to the nearest dollar as​ needed.)

Answer 1

Solution:-

Let X1 be income from teaching, X2 be income from selling, X3 be income from advertising.

Now total Income is X=X1+X2+X3. And

µ(X1)=40000, µ(X2)=25000, µ (X3)=18000

σ(X1)=4000^2, σ(X2)=5000^2, σ(X3)=6000^2

Hence,

µ(X)= µ(X1)+ µ(X2)+ µ(X3)

       = 40000 + 25000 + 18000

       = 83,000

Hence, the mean of total income from three sources is $83,000

Also,

σ(X1)=4000^2, σ(X2)=5000^2, σ(X3)=6000^2

σ(X)^2 = σ(X1) + σ(X2) + σ(X3)

            =4000^2 + 5000^2 + 6000^2

            =4,000,000 + 25,000,000 + 36,000,000

            =65,000,000

    σ(X) = √65,000,000

    σ(X) = 8062.26

Hence, the standard deviation of total income from three sources is $8,062.26