Question

A continuous random variable is uniformly distributed in the interval a<X<b the Lower quartile is 5 and upper quartile is 9 find
a. the values of a and b
b. p(6<X<7)
c. the cumulative distributive function

راز​

Answer 1

(a) X has a probabiliity density of

$f_X(x) = \begin{cases}\dfrac1{b-a}&\text{if }a$

If the lower quartile is 5 and the upper quartile is 9, then

$\displaystyle \int_a^5 f_X(x)\,\mathrm dx = 0.25 \text{ and } \int_9^b f_X(x)\,\mathrm dx = 0.25$

Computing the integrals gives the following system of equations:

$\displaystyle \int_a^5\frac{\mathrm dx}{b-a} = \frac{5-a}{b-a} = 0.25 \\\\ \int_9^b \frac{\mathrm dx}{b-a} = \frac{b-9}{b-a} = 0.25$

5 - a = 0.25 (b - a)   ==>   0.75a + 0.25b = 5   ==>   3a + b = 20

b - 9 = 0.25 (b - a)   ==>   0.25a + 0.75b = 9   ==>   a + 3b = 36

Eliminate a :

(3a + b) - 3 (a + 3b) = 20 - 3×36

-8b = -88

==>   b = 11   ==>   a = 3

and so P(X = x) = 1/(11 - 3) = 1/8 for all 3 < x < 11.

(b)

$\displaystyle P(6$

(c) The distribution function is then

$\displaystyle F_X(x) = \int_{-\infty}^x f_X(t)\,\mathrm dt = \begin{cases}0&\text{if }x\le3 \\ \dfrac x8 &\text{if }3$