Bernard Szlachta: /* Plotting Area Under normal distribution */

2016-03-09T03:03:04Z

Plotting Area Under normal distribution

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[[Category:Intro to R|01]]

== Probability Distributions ==
Each probability distribution will have four associated functions starting with d, p, q, r. Below normal distribution example
* '''dnorm''' - probability density function
* '''pnorm''' - density function (cumulative density)
* '''qnorm''' - quantile function
* '''rnorm''' - random varies

Other distributions will have similar functions, e.g. dt,pt,qt,rt, df, dbinom etc....

== Exercises ==
1. You flip a fair coin 10 times. What is the probability of getting 8 or more heads?
{{Show Answer|0.0546875}}
2. Assuming that the human height follows normal distribution with the mean of 174cm and standard deviation of 12cm, what proportion of goods a trouser manufacture should produce for people between 162 and 174cm?
{{Show Answer|Around 34%}}

== Summarizing Distribution ==
Parts of this tutorial is based on:
http://cran.r-project.org/doc/manuals/R-intro.pdf

> attach(faithful)
> summary(eruptions)
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.600 2.163 4.000 3.488 4.454 5.100

> fivenum(eruptions)
[1] 1.6000 2.1585 4.0000 4.4585 5.1000

> stem(eruptions)
The decimal point is 1 digit(s) to the left of the |
16 | 070355555588
18 | 000022233333335577777777888822335777888
20 | 00002223378800035778
22 | 0002335578023578
24 | 00228
26 | 23
28 | 080
30 | 7
32 | 2337
34 | 250077
36 | 0000823577
38 | 2333335582225577
40 | 0000003357788888002233555577778
42 | 03335555778800233333555577778
44 | 02222335557780000000023333357778888
46 | 0000233357700000023578
48 | 00000022335800333
50 | 0370

hist(eruptions)

hist(eruptions, seq(1.6, 5.2, 0.2), prob=TRUE)

lines(density(eruptions, bw=0.1))

rug(eruptions)

=== Empirical Cumulative Distribution ===
plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)

It seems there are two distributions (as two modes and histogram would suggest).
Let us try to split them.

long <- eruptions[eruptions > 3]
plot(ecdf(long), do.points=FALSE, verticals=TRUE)
x <- seq(3, 5.4, 0.01)

Let us fit normal distribution cumulative distribution function
lines(x, pnorm(x, mean=mean(long), sd=sd(long)), lty=3)

And closer look at Quantile-quantile (Q-Q) plot
par(pty="s")
qqnorm(long); qqline(long)

== Graphing Probability Distributions ==
Take example of calculating chances of getting 8 out of 10 heads.
plot(dbinom(seq(1,10),10,0.5),type="h")
:[[File:ClipCapIt-160309-062559.PNG|300px]]

old.par <- par(mfrow=c(1, 2))
plot(dbinom(seq(1,10),10,0.5),type="h")
plot(pbinom(seq(1,10),10,0.5),type="h",col=2)
par(old.par)

x <- seq(-4,4,length = 1000)
plot(x, dnorm(x),type="l")
:[[File:ClipCapIt-160309-064847.PNG|300px]]

curve(dt(x,4),-4,4,add = T,col=2)
:[[File:ClipCapIt-160309-070228.PNG|300px]]

== Plotting Area Under normal distribution ==
# Children's IQ scores are normally distributed with a
# mean of 100 and a standard deviation of 15. What
# proportion of children are expected to have an IQ between
# 80 and 120?

mean=100; sd=15
x <- seq(-4,4,length=100)*sd + mean
hx <- dnorm(x,mean,sd)
plot(x, hx,type="l")
i <- x >= 80 & x <= 120
polygon(c(80,x[i],120),
c(0,hx[i],0),
col="red")

# Orders are normally distributed (mean 100, sd=15). What proportion of are expected to have an value between 80 and 120?
:[[File:ClipCapIt-160309-071611.PNG]]

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Bernard Szlachta: /* Plotting Area Under normal distribution */