Chapter 7.1, Problem 12P

Expand Your knowledge: Continuous Uniform Probability Distribution

Let $\alpha$ and $\beta$ be any two constants such that $\alpha <\beta$. Suppose we choose a point x at random in the interval from a to 0. In this context, the phrase at random is taken to mean that the point x is as likely to be chosen from one particular part of the interval as any other part. Consider the rectangle.

The base of the rectangle has length $\alpha$ - $\beta$ and the height of the rectangle is $1/\left(\alpha \text{ -}\beta \right)$, so the area of the rectangle is 1. As such, this rectangle's top can be thought of as pan of a probability density curse. Since we specify that x

must lie between $\alpha$ and $\beta$ the probability of a point occurring outside the interval $\left[\alpha ,\beta \right]$ is, by definition. 0. From a geometric point of view, x chosen at random from $\alpha$ to $\beta$ means we are equally likely to land any where in the interval from $\alpha$ to $\beta$ For this reason, the lop of the (rectangle's) density curve is flat or uniform.

Now suppose that a and b are numbers such that $\alpha \le a<b\le \beta$. What is the probability that a number x chosen at random from $\alpha$ to $\alpha$ will fall in the interval [a, b]? Consider the graph.

Because x is chosen at random from [a, b], the area of the rectangle that lies above [a, b] is the probability that x lies in $\left[\alpha ,\beta \right]$ This area is

$P\left(a<x<b\right)=\frac{b-a}{\beta -\alpha }$

In this way. we can assign a probability to any interval inside |a. b|. This probability distribution is called the continuous uniform distribution (also called the rectangular distribution). Using some extra mathematics, it can be shown that if x is a random variable with this distribution, then the mean and standard deviation of x are

$\mu =\frac{\alpha +\beta }{2}\text{ and }\sigma =\frac{\beta -\alpha }{\sqrt{12}}$

Sedimentation experiments are very important in the study of biology, medicine, hydrodynamics, petroleum engineering, civil engineering, and so on. The size (diameter) of approximately spherical particles is important since larger particles hinder and sometimes Mock the movement of smaller particles. Usually the size of sediment particles follows a uniform distribution (Reference: Y. Zimmels, "Theory of Kindred Sedimentation of Poly disperse Mixtures.” AIChE Journal, Vol. 29. No. 4. pp. 669-676).

Suppose a veterinary science experiment injects very small, spherical pellets of low-level radiation directly into an animal’s bloodstream The purpose is to attempt to cure a form of recurring cancer. The pellets eventually dissolve and pass through the animal's system. Diameters of the pellets are uniformly distributed from 0.015 mm to 0.065 mm If a pellet enters an artery, what is the probability that it will be the following sizes?

(a) 0.050 mm or larger. Hint: All particles are between 0.015 mm and 0.065 mm, so larger than 0.050 means $0.050\le x\le \mathrm{0.065.}$

(b) 0.040 mm or smaller

(c) between 0.035 mm and 0.055 mm

(d) Compute the mean size of the particles.

(e) Compute the standard deviation of panicle size.