Question

The number of persons living per household in a city was collected and were summarized. The frequency table is given below: # people # households (millions 1 23 45 6 7 | 3.1 | 4.5 | 3.8 | 2.5 | 2.4 | 2.5 | 1.2 Define X as the number of people per household. (a) Write down the probability distribution of X. (b) Find P[X2 4] (c) Find P[2 S X< 4] (d) Find the mean, variance and standard deviation for number of people per household

Answer 1

a)
Total number of hosehold = 20

We need to find P(1) = 3.1/20 = 0.155

Similarly, calculated for other 6

x   P(x)
1   0.155
2   0.225
3   0.19
4   0.125
5   0.12
6   0.125
7   0.06

b)

P(x>=4) = P(x =4) + P(x =5) + P(x = 6) + P(x =7)
= 0.125 + 0.12 + 0.125 + 0.06
= 0.430

c)

P(2 < = x < 4)

P(x< 4) - P(x< =2)
= 0.19 + 0.225 + 0.155 - 0.225 - 0.155
= 0.19

d)

mean = sum of (x *(P(x))
= 1 * 0.155 + 2 * 0.225 + 3 * 0.19 + 4 * 0.125 + 5 * 0.12 + 6 * 0.125 + 7 * 0.06
= 3.445

variance = sum of (x - mean)^2/n-1
= (1-3.445)^2 * 0.155 + (2-3.445)^2 * 0.225 + (3-3.445)^2 * 0.19 + (4-3.445)^2 * 0.125 + (5-3.445)^2 * 0.12 + (6-3.445)^2 * 0.125 + (7-3.445)^2 * 0.06/6
= 0.5562

std.ev =sqrt(0.5562)
= 0.7458