Question

1. The table shows the number of computers per household in the United States.

Computers	0	1	2	3	4	5
Number of households (in millions)	27	47	24	10	4	2

1. Construct a probability distribution.
2. Graph the probability distribution using a histogram and describe its shape.
3. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.
4. Find the probability of randomly selecting a household that has at least four computers.

Answer 1

solution:

a) Probability Distribution

  

Computers	No.of households	P(X)	X*P(X)	X^2 * P(X)
0	27	27/114 = 0.2368	0	0
1	47	47/114=0.4123	0.4123	0.4123
2	24	24/114=0.2105	0.4210	0.8420
3	10	10/114=0.0877	0.2631	0.7893
4	4	4/114=0.0351	0.1404	0.5616
5	2	2/114=0.0175	0.0875	0.4375
	= 114		= 1.3243	=3.0425

c)

i) Mean = E[X] =

= 1.3243

Therefore, Mean = 1.3243

ii)Variance = = E[X^2] - E[X]^2

where E[X^2] =  

Therefore,Variance == = E[X^2] - E[X]^2 = 3.0425 - (1.3243)^2 = 1.2887

iii)Standard Deviation = =√ Variance = √ 1.2887 = 1.1352

Interpretation: The Expected value is 1.3243 .In the United States On Average every household has 1 computer.

The variance which is in square units (so we can't interpret it).Standard Deviation is 1.1352 which means you can expect 1computer to each household in US which is little vary about its mean.

d) P(Randomly selected household has four computers) = P(X>=4)

= P(X=4) + P(X=5)

= 0.0351+0.0175

= 0.0526

Probability of randomly selected household has four computers = 0.0526

b)