Question

An important factor in solid missile fuel is the
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) = 3
x
−4, x>
1,
0
,
elsewhere
.
(a) Verify that this is a valid density function.
(b) Evaluate F(x).
(c) What is the probability that a random particle
from the manufactured fuel exceeds 4 micrometers?

Answer 1

a)

$$f(x)=\begin{cases} 3x^{-4 },& x>1 \\ 0, &elsewhere \end{cases}$$

$f(x)\geq0,$ for all $x$

$$\displaystyle\int_{-\infin}^\infin f(x)dx=\displaystyle\int_{1}^{\infin}3x^{-4}dx=[-x^{-3}]\begin{matrix} \infin \\ 1 \end{matrix}$$

$$=-0+1=1$$

$f(x)\geq0,$ for all $x$

$\displaystyle\int_{-\infin}^\infin f(x)dx=1$

Therefore the function $f(x)$ is a valid density function.

b)

$$F(x)=\displaystyle\int_{-\infin}^x3t^{-4}dt=[-t^{-3}]\begin{matrix} x \\ 1 \end{matrix}=-x^{-3}+1$$

For $x>1$

$$F(x)=\displaystyle\int_{1}^xf(t)dt$$

$$F(x)=\begin{cases} 0, & x\leq1 \\ -x^{-4 }+1,& x>1 \end{cases}$$

c)

$$P(X>4)=\displaystyle\int_{4}^{\infin}f(t)dt=[-x^{-3}]\begin{matrix} \infin \\ 4 \end{matrix}$$

$$=-0+4^{-3}=\dfrac{1}{64}=0.015625$$

The probability that a random particle from the manufactured fuel exceeds 4 micrometers

is $0.015625.$