Question

Suppose that contamination particle size (in micrometers) canbe modeled as

Answer 1

Answer:

c) 2

d) 0.96

Step-by-step explanation:

We are given the following in the question:

$f(x) = 2x^{-3}, x > 1\\~~~~~~~= 0, x \leq 1$

a) probability density function.

$\displaystyle\int^{\infty}_{\infty}f(x) dx = 1\\\\\displaystyle\int^{\infty}_{-\infty}2x^{-3}dx = 1\\\\\displaystyle\int^{\infty}_{1}2x^{-3}dx\\\\\Rightarrow \big[-x^{-2}\big]^{\infty}_1\\\\\Rightarrow -(0-1) = 1$

Thus, it is a probability density function.

b) cumulative distribution function.

$P(X$

c) mean of the distribution

$\mu = \displaystyle\int^{\infty}_{-\infty}xf(x) dx\\\\\mu = \int^{\infty}_12x^{-2}dx\\\\\mu = (-\frac{2}{x})^{\infty}_{1}\\\\\mu = 2$

d) probability that the size of random particle will be less than 5 micrometers

$P(X$