Question

Please show all formulas so I can know how to work it out when it comes to test time please! Thank you! Given the following probability distributions for Austin Tech Inc. (ATI) and Stephen’s Industrial (SI) answer the following questions: (Please show all work with formulas)

State of Economy Probability ATI Returns SI Returns

Exceptional 20% 22% 37%

Average 50% 9% 6%

Recession 30% 1% -18%

A. What is the standard deviation for both ATI and SI?

B. Assuming a continuous distribution, approximately 95.44% of the time, the return for ATI will fall between _____ and _____.

C. What is the covariance between the returns of ATI and SI?

D. What is the correlation coefficient between ATI and SI?

Please show all formulas so I can know how to work it out when it comes to test time please! Thank you!

Answer 1

(A) standard deviation of ATI(x)=7.28

standard deviation of SI(y)=19.08

var(x)=E(x²)-E(x)*E(x)=137.6-9.2*9.2=52.96

sd(x)=sqrt(var(x))=sqrt(52.96)=7.28

E(x2)=sum(p*x²)=137.6 and E(x)=sum(x*p)=9.2

var(y)=389-5*5=364, sd(y)=sqrt(364)=19.08

(B) required confidence interval=mean(x) $\pm$ z(0.9544)*sd(x)=9.2 $\pm$ 2*7.28=9.2 $\pm$ 14.56=(-5.36, 23.76)

mean(x)=E(x)

(C)E(xy)=sum(p*x*y)=sum(0.2*22*37+0.5*9*6-0.3*1*18)=184.4

covariance(x,y)=E(xy)-E(x)E(y)=184.4-9.2*5=138.4

(D)correlation(x,y)=covariance(x,y)/(sqrt(var(x)*var(y))=138.4/sqrt(52.96*364)=0.9968

following information has been generated for the answering of above questions

state of economy	Probability(p)	ATI(x)	SI(y)	p*x	p*x2	p*y	p*y2
Exceptional	0.2	22	37	4.4	96.8	7.4	273.8
Average	0.5	9	6	4.5	40.5	3	18
Recession	0.3	1	-18	0.3	0.3	-5.4	97.2
sum	1	32	25	9.2	137.6	5	389