Question

( gileht mode or tu hings in your bags then place your bags in front. Wear your rovided, present your complete solutions and answers. Us Warning: Cheating in examinations is a major offense. Fou eadmission from the University. (Reference: Student Disci ourse Outcome 1 (100 pts) 1. An important factor in solid missile fuel is the |2. particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by f(x)-cx*, x>1 cx4, x >1 0, elsewhere Find c if f(x) is a density function. What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers? a) b) olution:

Answer 1

a)

If f(x) is a density function then,

$\int_x f(x)dx =1$

$=>\int_{x=1}^{x=\infty} cx^{-4}dx =1$

$=>c\int_{x=1}^{x=\infty} x^{-4}dx =1$

Integrating we get :

$=>c\left [ \frac{x^{-4+1}}{-4+1} \right ]_1^{\infty} =1$

$=>c\left [ \frac{x^{-3}}{-3} \right ]_1^{\infty} =1$

$=>c\left [ \frac{\infty^{-3}}{-3}-\frac{1^{-3}}{-3} \right ]=1$

$=>\frac{c}{3}=1$

$=>c=3$

b)

Probability that a random particle from the manufactured fuel exceeds 4 micrometers = P(X>4)

$P(X>4) = \int_{x=4}^{x=\infty}f(x)dx$

$= \int_{x=4}^{x=\infty}cx^{-4}dx$

$= \int_{x=4}^{x=\infty}3x^{-4}dx$

Integrating we get :

$= 3\left [ \frac{x^{-4+1}}{-4+1} \right ]_{4}^{\infty}$

$= 3\left [ \frac{x^{-3}}{-3} \right ]_{4}^{\infty}$

$= 3\left [ \frac{{\infty}^{-3}}{-3}-\frac{{4}^{-3}}{-3} \right ]$

$=4^{-3} = 0.0156$

Probability that a random particle from the manufactured fuel exceeds 4 micrometers = 0.0156