# 5.2 The uniform distribution

 Page 1 / 4
Continuous Random Variable: Uniform Distribution is part of the collection col10555 written by Barbara Illowsky and Susan Dean. It describes the properties of the Uniform Distribution with contributions from Roberta Bloom.

The previous problem is an example of the uniform probability distribution .

Illustrate the uniform distribution . The data that follows are 55 smiling times, in seconds, of an eight-week old baby.

 10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9 12.8 14.8 22.8 20 15.9 16.3 13.4 17.1 14.5 19 22.8 1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9 19.2 9.8 5.8 6.9 2.6 5.8 21.7 11.8 3.4 2.1 4.5 6.3 10.7 8.9 9.4 9.4 7.6 10 3.3 6.7 7.8 11.6 13.8 18.6

sample mean = 11.49 and sample standard deviation = 6.23

We will assume that the smiling times, in seconds, follow a uniform distribution between 0 and 23 seconds, inclusive. This means that any smiling time from 0 to and including 23 secondsis equally likely . The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.

Let $X$ = length, in seconds, of an eight-week old baby's smile.

The notation for the uniform distribution is

$X$ ~ $U\left(\mathrm{a,}b\right)$ where $a$ = the lowest value of $x$ and $b$ = the highest value of $x$ .

The probability density function is $f(x)=\frac{1}{b-a}$ for $((a, x), b)$ .

For this example, $x$ ~ $U\left(0,23\right)$ and $f(x)=\frac{1}{23-0}$ for $((0, x), 23)$ .

Formulas for the theoretical mean and standard deviation are

$(\mu , \frac{a+b}{2})$ and $(\sigma , \sqrt{\frac{\left(b-a{\right)}^{2}}{12}})$

For this problem, the theoretical mean and standard deviation are

$((\mu , \frac{0+23}{2}), 11.50)$ seconds and $((\sigma , \sqrt{\frac{\left(\mathrm{23}-0{\right)}^{2}}{12}}), 6.64)$ seconds

Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation.

What is the probability that a randomly chosen eight-week old baby smiles between 2 and 18 seconds?

Find $((P\left(2, x), 18\right))$ .

$(((((P\left(2, x), 18\right)), \left(\text{base}\right)\left(\text{height}\right)), \left(18-2\right)\cdot \frac{1}{23}), \frac{16}{23})$ . Find the 90th percentile for an eight week old baby's smiling time.

Ninety percent of the smiling times fall below the 90th percentile, $k$ , so $((P\left(x, k\right)), 0.90)$

$((P\left(x, k\right)), 0.90)$

$\left(\text{base}\right)\left(\text{height}\right)=0.90$

$\left(k-0\right)\cdot \frac{1}{23}=0.90$

$((k, 23\cdot 0.90), 20.7)$ Find the probability that a random eight week old baby smiles more than 12 seconds KNOWING that the baby smiles MORE THAN 8 SECONDS .

Find $((P\left(x, 12|x), 8\right))$ There are two ways to do the problem. For the first way , use the fact that this is a conditional and changes the sample space. The graph illustrates the new sample space. You already know the baby smiled morethan 8 seconds.

Write a new $f(x)$ : $((f(x), \frac{1}{23-8}), \frac{1}{15})$

for $((8, x), 23)$

$((((P\left(x, 12|x), 8\right)), \left(23-12\right)\cdot \frac{1}{15}), \frac{11}{15})$ For the second way, use the conditional formula from Probability Topics with the original distribution $X$ ~ $U\left(0,23\right)$ :

$P\left(A|B\right)=\frac{P\left(A\phantom{\rule{2pt}{0ex}}\text{AND}\phantom{\rule{2pt}{0ex}}B\right)}{P\left(B\right)}$ For this problem, $A$ is $(\left(x, 12\right))$ and $B$ is $(\left(x, 8\right))$ .

So, $((((((P\left(x, 12|x), 8\right)), \frac{\left(x>12\phantom{\rule{2pt}{0ex}}\text{AND}\phantom{\rule{2pt}{0ex}}x>8\right)}{(P\left(x, 8\right))}), \frac{(P\left(x, 12\right))}{(P\left(x, 8\right))}), \frac{\frac{11}{23}}{\frac{15}{23}}), 0.733)$ Uniform : The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 15 minutes, inclusive.

What is the probability that a person waits fewer than 12.5 minutes?

Let $X$ = the number of minutes a person must wait for a bus. $a$ = 0 and $b$ = 15. $x~U\left(0,15\right)$ . Write the probability density function. $f(x)=\frac{1}{15-0}=\frac{1}{15}$ for $((0, x), 15)$ .

Find $(P\left(x, 12.5\right))$ . Draw a graph.

$((P\left(x, k\right)), \left(\text{base}\right)\left(\text{height}\right))=\left(12.5-0\right)\cdot \frac{1}{15}=0.8333$

The probability a person waits less than 12.5 minutes is 0.8333. On the average, how long must a person wait?

Find the mean, $\mu$ , and the standard deviation, $\sigma$ .

$\mu =\frac{a+b}{2}=\frac{15+0}{2}=7.5$ . On the average, a person must wait 7.5 minutes.

$\sigma =\sqrt{\frac{\left(b-a{\right)}^{2}}{12}}=\sqrt{\frac{\left(\mathrm{15}-0{\right)}^{2}}{12}}=4.3$ . The Standard deviation is 4.3 minutes.

Ninety percent of the time, the time a person must wait falls below what value?

This asks for the 90th percentile.

Find the 90th percentile. Draw a graph. Let $k$ = the 90th percentile.

$((P\left(x, k\right)), \left(\text{base}\right)\left(\text{height}\right))=\left(k-0\right)\cdot \left(\frac{1}{15}\right)$

$0.90=k\cdot \frac{1}{15}$

$\mathrm{k = \left(0.90\right)\left(15\right) = 13.5}$

$k$ is sometimes called a critical value.

The 90th percentile is 13.5 minutes. Ninety percent of the time, a person must wait at most 13.5 minutes. Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid. By Cath Yu By Saylor Foundation By Cath Yu By OpenStax By OpenStax By Keyaira Braxton By OpenStax By Rachel Woolard By Richley Crapo By David Bourgeois