Calculate the expected monthly laptop computer sales

Question

The below information contains the probabilities of various combinations of monthly sales volumes of Dell (X) and HP (Y) laptop computers from online sites. Are the monthly sales of these two competitors independent of each other? Explain your answer.

		HP sales (y)
		25	30	35	40	45
Dell Sales (X)	25	0.01	0.01	0.01	0.02	0.01	0.06
	30	0.02	0.03	0.03	0.05	0.01	0.14
	35	0.04	0.04	0.04	0.08	0.03	0.23
	40	0.04	0.04	0.04	0.08	0.03	0.23
	45	0.02	0.03	0.03	0.05	0.01	0.14
	50	0.01	0.01	0.01	0.02	0.01	0.06
		0.14	0.16	0.16	0.3	0.1	0.86

a) Find the marginal distributions of X and Y. Interpret your findings.

	X	P(X)	Xi2 x Pi
Dell Sales (X)	25	0.06	37.5
	30	0.14	126
	35	0.23	281.75
	40	0.23	368
	45	0.14	283.5
	50	0.06	150
sum		0.86	1246.75

 

	HP sales (y)					sum
Y	25	30	35	40	45	175
P(Y)	0.14	0.16	0.16	0.3	0.1	0.86
Yi2 x P(yi)	87.5	144	196	480	202.5	1110

X and Y represent the sales of Dell and HP laptop computers in 1000s of units. As seen in the spreadsheet, the marginal distributions indicate that intermediate sales value of both the laptops are most likely where extreme sales value are less likely.

b) Calculate the expected monthly laptop computer sales volumes for Dell and HP at these sites.

E(X) = _{i = 1}ΣⁿX_i x P(X_i)

E(X) =
E(Y) =

c) Calculate the standard deviations of the monthly laptop computer sales volumes for Dell and HP at these sites.
VAR = Sum(X_i² x P_i)) - mean²

VAR(x)=
VAR(Y)

SD(X)
SD(Y)

d) Find and interpret the conditional distribution of X given Y

conditional distribution of X given Y:

		HP sales (y)
		25	30	35	40	45
Dell Sales (X)	25	0.07	0.06	0.06	0.07	0.10
	30	0.14	0.19	0.19	0.17	0.10
	35	0.29	0.25	0.25	0.27	0.30
	40	0.29	0.25	0.25	0.27	0.30
	45	0.14	0.19	0.19	0.17	0.10
	50	0.07	0.06	0.06	0.07	0.10
sum		1	1	1	1	1

e) Find and interpret the conditional distribution of Y given X.

conditional distribution of Y given X:

 

		HP sales (y)
		25	30	35	40	45	SUM
Dell Sales (X)	25	0.17	0.17	0.17	0.33	0.17	1
	30	0.14	0.21	0.21	0.36	0.07	1
	35	0.17	0.17	0.17	0.35	0.13	1
	40	0.17	0.17	0.17	0.35	0.13	1
	45	0.14	0.21	0.21	0.36	0.07	1
	50	0.17	0.17	0.17	0.33	0.17	1

f)  Find and interpret the correlation between X and Y. Are these random variables independent

COV(X,Y) =

		HP sales (y)
		25	30	35	40	45
Dell Sales (X)	25	6.25	7.5	8.75	20	11.25	53.75
	30	15	27	31.5	60	13.5	147
	35	35	42	49	112	47.25	285.25
	40	40	48	56	128	54	326
	45	22.5	40.5	47.25	90	20.25	220.5
	50	12.5	15	17.5	40	22.5	107.5
		131.25	180	210	450	168.75	1140

E(XY ) =
COV(XY) =

Corr = COV(XY)/SQRT{V(X) x V(Y)}

There is a strong lienar relationship between X and Y. That is as the value of X increases, the value of Y also increases.

Albright, S. Christian; Winston, Wayne; Zappe, Christopher (2010-10-12). Data Analysis and Decision Making (Page 192). Cengage Textbook. Kindle Edition.