Question

# A random variable X has the following discrete probability distribution. x 12 19 22 24 27...

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).

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

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