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

(a) X has a probabiliity density of

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

Computing the integrals gives the following system of equations:

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)

(c) The distribution function is then