Question

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate Standard deviation of X.

Answer 1

We know that Standard deviation (SD) = `sqrt("Variance")`

Variance= E(X²) – [E(X)]²

E(X²) = `1 xx 2/50 + 4 xx 3/50 + 9 xx 4/50 + 16 xx 5/50 + 25 xx 10/50 + 36 xx 12/50 + 49 xx 14/50`

= `2/50  + 12/50 + 36/50 + 80/50 + 250/50 + 432/50 + 686/50`

= `1498/50`

∴ Variance (X) = `1498/50 - (26/5)^2`

= `1498/50 - 676/25`

= `(1498 - 1352)/50`

= `146/50`

= 2.92

∴ S.D = `sqrt(2.92)`

= 1.7  .....(Approx)