Question

A random variable X has the following discrete probability distribution.

x	12	19	22	24	27	32
p(x)	0.13	0.25	0.18	0.17	0.11	0.16

Calculate σ = standard deviation of X (up to 2 decimal places).

Answer 1

Solution:

Given in the question
that data is discrete data
For standard deviation, first we need to calculate Mean

X	P(X)	X.P(X)
12	0.13	1.56
19	0.25	4.75
22	0.18	3.96
24	0.17	4.08
27	0.11	2.97
32	0.16	5.12

Mean = 1.56+4.75+3.96+4.08+2.97+5.12 = 22.44

Standard deviation can be calculated as
SD = sqrt(summation(Xi-mean)^2 *P(Xi))

X	P(X)	X.P(X)	Xi-Mean	(Xi-mean)^2	(Xi-mean)^2*P(X)
12	0.13	1.56	-10.44	108.9936	14.169168
19	0.25	4.75	-3.44	11.8336	2.9584
22	0.18	3.96	-0.44	0.1936	0.034848
24	0.17	4.08	1.56	2.4336	0.413711999999999
27	0.11	2.97	4.56	20.7936	2.287296
32	0.16	5.12	9.56	91.3936	14.622976

SD = sqrt(14.169+2.9584+0.0348+0.4137+2.287+14.62) = sqrt(34.4864) = 5.87