Question

A normal variable X has the probability density function as,

f(x) = constant × \(e^{−\frac{1}{50}(x−10)^2}\), – ∞ < x < ∞

Find the first quartile from this information.

Answer 1

f(x) = constant × \(e^{−\frac{1}{50}(x−10)^2}\), – ∞ < x < ∞

= constant ×\( e^{−\frac{1}{2}(\frac{x−10}5)^2}\)

Hence, µ = 10, σ = 5

First Quartile Q₁:

25 % of observations of the distribution is less than Q₁.

P[Z ≤ z₁] = 0.25
= P[- ∞ < Z ≤ 0] – P[z₁ ≤ Z ≤ 0]
= P[0 ≤ Z < ∞] – P[0 ≤ Z ≤ z₁]
∴ P[0 ≤ Z ≤ z₁] = 0.5000 – 0.2500
= 0.2500

From the table, corresponding to probability 0.2486, the value of z₁ = -0.67 and corresponding to probability 0.2518, the value of z₁ = – 0.68
∴ For probability 0.25