605-Discussion 5

Chapter 2 : Section 2.2 - Exercise 2 - Suppose you choose a real number X from the interval [2, 10] with a density function of the form where C is a constant.

                                    f(x) = Cx

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

if f(x) = Cx

\[ y = \int_{2}^{10} Cx dx = (Cx^2)/2 + K[2, 10] = 50C - 2C = 48C \]

=> y must be <= 1, since this is the area under the probability density function from 2 to 10

=> $48C = 1$ => $C = 1/48 $

\[ P(E) = \int_{a}^{b} (x/48) dx = (b^2 - a^2)/96 \]

\[ P(X > 5) = \int_{5}^{10} (x/48) dx = (10^2 - 5^2)/96 = 75/96 \]

\[ P(X < 7) = \int_{2}^{7} (x/48) dx = (7^2 - 2^2)/96 = 45/96 \]

$y = x^2 - 12x + 35 = (x - 7)(x - 5)$ => for y to be greater than 0 then x must be >= 7 or <= 5

Therefore =>

\[ P(x^2 - 12x + 35 > 0) = P(x <= 5 or x >= 7) = \int_{2}^{5} x/48 dx + \int_{7}^{10} x/48 dx = ((5^2 - 2^2) + (10^2 - 7^2))/96 = (21 + 51)/96 = 72/96 = 3/4 \]

605-Discussion 5

Ohannes (Hovig) Ohannessian

9/26/2018

Chapter 2 : Section 2.2 - Exercise 2 - Suppose you choose a real number X from the interval [2, 10] with a density function of the form where C is a constant.