The number of fans for a movie depends on how many star actors make an appearanc

Question

The number of fans for a movie depends on how many star actors make an appearance on screen and the size of the budget. Suppose that the number of fans equals 5 + 3S + .02B where S is the number of star actors and B is the size of the budget. The following represents the joint distribution of S and B in the relevant movies that we care to examine:

a) Fill in the missing values of the table. b) Make a new table with the values of the conditional PDFs of S. (That is, there will be two PDFs, one for P(S|B = 100) and P(S|B = 200).) c) Using b), calculate the expected values of S, conditonal on B. Compare these two expectations. Does the relative magnitude of the expected values yield any intuition about how the two variables are related? d) Calculate the E[B], E[S], Var[B], and Var[S]. e) Next calculate the covariance of B and S. Finally, calculate the correlation coefficient. f) If a movie is selected at random, what is the expected number of fans? What is the variance of the number of fans?

B=200 P(S=s)| 0.2 0.25 0.6 0.3

Explanation / Answer

a)

b)

here each individual cell represent conditional probability:

c) E(S|100) =1*0.5+3*0.3750+5*0.1250=2.25

E(S|200)=1*0.1667+3*0.5+5*.3333 =3.3333

from abvoe it can be seen that as budget increaes; number of star actors also increase. therefore two variables are positively related

d) from above marginal pmf of B:

and marginal pmf of S :

therefore E(B) =160 ; E(S) =2.9 ; Var(B) =2400 ; Var(S) =2.19

e) here E(XY) =490

therefore Cov(B,S) =E(XY)-E(X)*E(Y) =26

correlation coefficient = Cov(B,S)/(Var(B)*Var(S))1/2 =0.3586

f)

expected number of fans =5+3E(S)+0.02E(B) =5+3*2.9+0.02*160=16.9

variance =32*Var(S)+0.022*Var(B)+2*3*0.02*Cov(S,B) =23.79

S B 1 3 5 Total 100 0.2000 0.1500 0.0500 0.4000 200 0.1000 0.3000 0.2000 0.6000 Total 0.3000 0.4500 0.2500 1.0000

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