Question

5. The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variable with density cr, for 0<a<3, f(x)-c(6-z) for 3<x < 6, 0 otherwise () Find the value of c which makes f a probability density function (i) What is the probability that the number of kilos of bread that will be sold in a day is, a) more than 300 kilos? b) between 150 and 450 kilos? (i) Denote by A and B the events in (a) and (b), respectively. Are A and B independent?

Answer 1

$5.~i.~Since~f(x)~is~a~pdf~so\\ c\int_0^3xdx&plus;c\int_3^6(6-x)dx=1~or,~c\left[\frac{x^2}{2} \right ]^3_0&plus;c\left[6x-\frac{x^2}{2} \right ]^6_3=1\\ or,~36c=1~or,~c=\frac{1}{36}\\ ii.~a.~Probability~that~the~number~of~kilos~of~bread~that~will~be~sold~in~a\\ day~is~more~than~300~kilos=P(X>3)=\frac{1}{36}\int_3^6(6-x)dx=\frac{7}{8}\\ b.~Probability~that~the~number~of~kilos~of~bread~that~will~be~sold~in~a\\ day~is~between~150~and~450~kilos=P(1.5<X<4.5)\\ =P(1.5<X<3)&plus;P(3<X<4.5)\\ =\frac{1}{36}\int_{1.5}^3xdx&plus;\frac{1}{36}\int_3^{4.5}(6-x)dx=\frac{1}{36}\left[\frac{x^2}{2} \right ]^3_{1.5}&plus;\frac{1}{36}\left[6x-\frac{x^2}{2} \right ]^{4.5}_{3}=\frac{3}{16}\\ (iii)~A~and~B~are~not~independent~since\\ P(A\cap B)=P(1.5<X<3)=\frac{1}{36}\int_{1.5}^3xdx=\frac{3}{32}$