Statistics 101 Review Example on Bivariate November 8, 2004A company makes two calculators: a scientific calculator and a business calculator.Let X = Monthly demand (in thousands) for the business calculators.Let Y = Monthly demand (in thousands) for the scientific calculators.We are given the following joint probability distribution function: x 1 2 3 4 1 .2 .1 .1 0 y 2 0 .1 .1 0 3 0 .1 .1 .2 Business Calculators Scientific CalculatorsSelling price $30 $50Manufacturing cost $20 $30Total Fixed cost $50,000 per month Questions1. a) Find p(x). Use p(x) to find the expected value and variance of X. b) Find p(y). Use p(y) to find the expected value and variance of Y.2. If we sell 1000 scientific calculators, find the probability distribution,expected value and variance for the number of business calculators sold. 3. Let Z be monthly profit (in $10,000 units) from both types of calculators.a) Express Z in terms of X and Y. b) Find p(z). Use p(z) to find the expected value and variance of Z. c) Find Cov(X,Y) and Corr(X,Y). Interpret the correlation in context. d) Use the rules for E(Z) and V(Z) to verify your answer in part b).4. a) What is the probability that total monthly profit exceeds $5,000?b) What is the probability that total monthly profit is negative?Solutions1. x p(x) xp(x) (x-)2p(x) y p(y) yp(y) (y-)2p(y) -- ----- ------ ------------ -- ----- ------- ------------ 1 .2 .2 (2.25) .2=.45 1 .4 .4 (1) .4=.4 2 .3 .6 (.25) .3=.075 2 .2 .4 (0) .2= 0 3 .3 .9 (.25) .3=.075 3 .4 1.2 (1) .4=.4 4 .2 .8 (2.25) .2=.45 --------------------------------------------------------------------------------------------- E(X) = 2.5 V(X) = 1.05 E(Y)=2.0 V(Y) = .82. x p(x|y=1) xp(x|y=1) (x)2p(x|y=1) -- ---------- ------------ ------------------- 1 .2/.4 =.5 .5 (1) .5 =.5 2 .1/.4=.25 .5 (4) .25= 1.0 3 .1/.4=.25 .75 (9) .25= 2.25 E(X|y=1) = 1.75 E(X2|y=1) = 3.75 V(X|y=1) = 3.75-(1.75)2 = .68753. a) Z = X +2Y-5 b) Table of values of Z x 1 2 3 4 z p(z) zp(z) (z-)2p(z) ----------------------------------- -- ---- ----- ------------ 1 -2 -1 0 1 -2 .2 -.4 (12.25).2=2.45y 2 0 1 2 3 -1 .1 -.1 (6.25).1= .625 3 2 3 4 5 0 .1 0 (2.25).1= .225 1 .1 .1 (.25) .1= .025 2 .1 .2 (.25) .1 = .025 3 .1 .3 (2.25) .1 =.225 4 .1 .4 (6.25) .1 = .625 5 .2 1.0 (12.25) .2=2.45 ------ --------------- E(Z)=1.5 V(Z)= 6.65c) E(XY)= (1*1).2+(2*1).1+(3*1).1+ (2*2).1 +(3*2) .1 + (2*3) .1 +(3*3) .1 +(4*3) .2 = 5.6 COV(X,Y) =E(XY)-E(X)E(Y)=5.6-2.5(2)=.6 CORR(X,Y)= .6/[1.05*.8]1/2 =.6547 The demand for the two calculators tend to go in the same direction. This is a relatively strong (.65 on a 0 to 1 scale) linear relationship.d) E(Z) = E(X+2Y-5)=E(X) +2E(Y) – 5 = 2.5 +2(2) – 5 = 1.5V(Z) =V(X+2Y-5)= V(X) +4V(Y)+2(1)(2) COV(X,Y) = 1.05+3.2+4(.6)= 6.654. a) Profit is greater than .5 ($5,000). Just add up prob. of cases = .1 +.1 +.1+.1+.2 = .6b) Profit is negative only when x=1 and y=1 or x=2 and y=1