Rprobcontinuous.Rmd

---
title: "Continuous Probability Distributions with R"
author: "Marianne Huebner, Michigan State University"
date: '`r Sys.Date()`'
output: word_document
---

### Normal distribution
Average height of US males are approximately normal distributed with mean 69.1 inches 
and standard deviation 2.9 inches. 


*Cumulative distribution function:* probability that a man is less than 70 inches tall.
```{r}
pnorm(70, mean=69.1, sd=2.9)

# What is the probability of a man being taller than 70 inches?
```{r}
1-pnorm(70, mean=69.1, sd=2.9)


# Calculate P(66< X <72) for X ~ Normal(69.9, 2.9)
```{r}
pnorm(72, mean=69.1, sd=2.9)-pnorm(66, mean=69.1, sd=2.9)
```

*Inverse cumulative distribution function.*
```{r}
qnorm(0.9,mean=69.1, sd=2.9)
```
There is a 90% chance that a man is `r qnorm(0.9,mean=69.1, sd=2.9)` inches tall or less.

Do the data come from a normal probability distribution?

```{r, fig.height=3, fig.width=5}
# generate normally distributed data
z<-rnorm(50)
# generate random numbers from a skewed distribution
y<-rexp(50, rate=1/3)

par(mfrow=c(1,2))
qqnorm(z, main="Normal distribution")
qqline(z)

qqnorm(y, main="Exponential distribution")
qqline(y)
par(mfrow=c(1,1))
```


The fill weight of baby food jars follows a normal distribution with mean 137.6 grams
and standard deviation of 1.6 grams. The label weight  on the jars is 135.0 grams. 
*Reference: J. Fisher. Computer assisted net weight control. Quality Progress, June 1983.*

```{r, fig.height=3, fig.width=4}
mean=137.6; sd=1.6
lb=100; ub=135

# draw a density curve for this normal distribution
x <- seq(-4,4,length=100)*sd + mean
px <- dnorm(x,mean,sd)

plot(x, px, type="n", xlab="fill weight of jars", ylab="", axes=FALSE)

# shade the area for which the probability is calculated
i <- x >= lb & x <= ub
lines(x, px)
polygon(c(lb,x[i],ub), c(0,px[i],0), col="gray") 

#Calculate the probability P(W<135) and paste it into the figure

area <- pnorm(ub, mean, sd)
result <- paste("P(W <",ub,") =", signif(area, digits=3))
mtext(result,3)
# add axes ticks and labels
axis(1, at=seq(130, 145, 2), pos=0)
```

Create a figure and shade it for the probability $P(134< W<136)$ 
```{r, fig.height=3, fig.width=4}
lb=134; ub=136

plot(x, px, type="n", xlab="fill weight of jars", ylab="", axes=FALSE)

# shade the area for which the probability is calculated
i <- x >= lb & x <= ub
lines(x, px)
polygon(c(lb,x[i],ub), c(0,px[i],0), col="gray") 


area <- pnorm(ub, mean, sd) - pnorm(lb, mean, sd)
result <- paste("P(",lb,"< W <",ub,") =", signif(area, digits=3))
mtext(result,3)
# add axes ticks and labels
axis(1, at=seq(130, 145, 2), pos=0)
```

Unknown mean: create a figure and shade it for the probability $P(W<135) = 0.01$
```{r, fig.height=3, fig.width=4}
sd=1.6
mean=135-qnorm(0.01)*sd
lb=100; ub=135

plot(x, px, type="n", xlab="fill weight of jars", ylab="", axes=FALSE)

# shade the area for which the probability is calculated
i <- x >= lb & x <= ub
lines(x, px)
polygon(c(lb,x[i],ub), c(0,px[i],0), col="gray") 


area <- pnorm(ub, mean, sd) 
result <- paste("P(W <",ub,") =",  signif(area, digits=3), "for mean=", round(mean,2))
mtext(result,3)
# add axes ticks and labels
axis(1, at=seq(130, 145, 5), pos=0)
```


### Exponential distribution


Sixty-watt light bulbs have an average life of 1000 hours.
The probability distribution of the lifetime of such a light bulb is exponential with rate=0.001 


*Cumulative distribution function:* probability that the lifetime is less than 100 hours.
```{r}
pexp(100, rate=1/1000)
```

What is the probability that the light bulb lasts at least 900 hours?
```{r}
1-pexp(900, 0.001)
```


Calculate $P(900< T <1200)$ for $T \tilde Exp(0.001)$
```{r}
pexp(1200, rate=0.001)-pexp(900, rate=0.001)
```


*Inverse cumulative distribution function.*
```{r}
qexp(0.5,rate=0.001)
```
The median lifetime of such  a bulb is `r qexp(0.5,rate=0.001)` hours.


The top 20% of the light bulbs last at least how many hours?
```{r}
qexp(0.8,rate=0.001)
```