Continuous density functions Problem 2 page 71

Suppose you choose a real number X from the interval [2,10] with a density function of the form f(x)=Cx where C is a constant

a)Find C. b) Find P(E) where E=[a,b] is a subinterval of [2,10]. c) Find P(X>5) , P(X<7), and P(X2−12X+35>0)

Solution:

  1. To find C, we require the probability over the entire sample space to sum to one.

\[\int_{2}^{10} f(x)dx=1\]

\[\int_{2}^{10} C(x)dx=1\] \[\int_{2}^{10} C(x)dx=\frac{C}{2}(10^2-2^2)=\frac{C}{2}(100-4)=1\] Therefore \[C=\frac{2}{96}=\frac{1}{48}\]

\[P(E)=\int_{a}^{b} f(x)dx=\frac{C}{2}(b^2-a^2)=\frac{1}{96}(b^2-a^2)\]

  1. Using the formula in part b

\[P(X>5)=P([5,10])=\frac{1}{96}(100-25)=\frac{75}{96}=\frac{25}{32}\] \[P(X<7)=P([2,7])=\frac{1}{96}(49−4)=\frac{45}{96}=\frac{15}{32}\] \[P(X^2-12X+35>0)=P((X−7)(X−5)>0)=P(X>7)+P(X<5)=(1−P(X<7))+(1−P(X>5))\]

\[P(X^2-12X+35>0)=(1−\frac{15}{32})+(1−\frac{25}{32})=\frac{24}{32}=34\]