```{r setup, include=FALSE}
knitr::opts_chunk$set(echo=TRUE, fig.align="center")
options(width=110)
```

#### Statistics for Laboratory Scientists ( 140.615 )

## Random Variables                                       

### The Binomial distribution

The example from class: the number of red balls in 9 draws with replacement from an urn for which the proportion of red balls is 20%.

##### The Binomial coefficient

One red ball.

```{r}
choose(9,1)
```

Two red balls.

```{r}
choose(9,2)
factorial(9)
factorial(2)
factorial(9-2)
factorial(9)/(factorial(2)*factorial(9-2))
```

All possibilities.

```{r}
choose(9,0:9)
```

##### The outcome probabilities

No red balls.
 
```{r}
(1-0.2)^9
choose(9,0)*(0.2^0)*(0.8^9)
dbinom(0,9,0.2)
```

All red balls.

```{r}
0.2^9
choose(9,9)*(0.2^9)*(0.8^0)
dbinom(9,9,0.2)
```

One red ball.

```{r}
choose(9,1)*(0.2^1)*(0.8^8)
dbinom(1,9,0.2)
```

All possibilities.

```{r}
dbinom(0:9,9,0.2)
barplot(dbinom(0:9,9,0.2),names=0:9)
```

##### The cumulative distribution function

```{r}
dbinom(0:4,9,0.2)
sum(dbinom(0:4,9,0.2))
pbinom(4,9,0.2)
1- pbinom(4,9,0.2)
pbinom(4,9,0.2,lower=FALSE)
```

##### Simulate Binomial outcomes

```{r}
set.seed(4)
rbinom(1,9,0.2)
rbinom(10,9,0.2)
x <- rbinom(100,9,0.2)
table(x)
prop.table(table(x))
table(x)/length(x)
round(dbinom(0:9,9,0.2),2)
```

### Poisson distribution

##### The probability function

```{r}
dpois(3,5)
exp(-5)*(5^3)/factorial(3)
dpois(0:15,5)
barplot(dpois(0:15,5),names=0:15)
```

##### The cumulative distribution function

```{r}
dpois(0:7,5)
sum(dpois(0:7,5))
ppois(7,5)
1 - ppois(7,5)
ppois(7,5,lower=FALSE)
```

##### Simulate Poisson outcomes

```{r}
rpois(1,5)
rpois(20,5)
x <- rpois(100,5)
table(x)
prop.table(table(x))
round(dpois(0:10,5),2)
```

### Poisson approximation for the Binomial

```{r}
n <- 50000
p <- 1/100000
lambda <- n*p
lambda
dbinom(0:5,n,p)
dpois(0:5,lambda)
dbinom(0:5,n,p) - dpois(0:5,lambda)
```

### The Gaussian distribution

##### The N(0,1) density function

```{r}
curve(dnorm(x), from=-3, to=3)
curve(dnorm(x), from=-3, to=3, ylab="f(x)", main="Standard Gaussian")
dnorm(0)
dnorm(1)
dnorm(3)
```

##### The N(0,1) cumulative distribution function

```{r}
curve(pnorm(x), from=-3, to=3, ylab="P(X < x)", main="CDF")
pnorm(0)
pnorm(2)
pnorm(-2)
pnorm(2) - pnorm(-2)
```

##### The N(0,1) inverse cumulative distribution function

```{r}
curve(qnorm(x), from=0, to=1, main="Inverse CDF", xlab="p", ylab="qnorm(p)")
qnorm(0.5)
qnorm(0.95)
qnorm(0.975)
qnorm(0.025)
```

##### Area under the curve

Highlight 95% of the area under the standard Gaussian N(0,1).

```{r}
L <- qnorm(0.025)
L
U <- qnorm(0.975)
U
curve(dnorm(x), from=-4, to=4)
abline(h=0)
abline(v=c(L,U), col="red", lty="dotted")
```

##### Simulation

Simulate some standard Gaussian data, plot the histogram, and superimpose the density.

```{r}
rnorm(10)
hist(rnorm(100), breaks=11)
y <- rnorm(10000)
hist(y, breaks=seq(-4.5,4.5,by=0.25))
hist(y, breaks=seq(-4.5,4.5,by=0.25), prob=TRUE)
curve(dnorm(x), from=-4.5, to=4.5, add=TRUE, col="red", lwd=2)
```

##### The Normal with a mean of 5 and standard deviation of 2

```{r}
curve(dnorm(x,mean=5,sd=2), from=-2, to=12, ylab="f(x)", main="Normal(mean=5,sd=2)")
curve(pnorm(x,mean=5,sd=2), from=-2, to=12, ylab="P(X < x)", main="CDF")
curve(qnorm(x,mean=5,sd=2), from=0, to=1, main="Inverse CDF", xlab="p", ylab="qnorm(p)")
```

##### Examples from class

```{r}
curve(dnorm(x,mean=69,sd=3), from=57, to=81, ylab="f(x)", main="Normal(mean=69,sd=3)")
```

What proportion of men are taller than 5’7”?

```{r}
pnorm(67,mean=69,sd=3,lower=FALSE)
pnorm(67,69,3,lower=FALSE)
1 - pnorm(67,69,3)
pnorm((67-69)/3,lower=FALSE)
pnorm(2/3)
```

What proportion of men are between 5’3” and 6’?

```{r}
pnorm(72,69,3)
pnorm(63,69,3)
pnorm(72,69,3) - pnorm(63,69,3)
pnorm((72-69)/3) - pnorm((63-69)/3)
pnorm(1) - pnorm(-2)
```

### The Uniform distribution

##### The Uniform (0,1) density function

```{r}
curve(dunif(x), from=-1, to=2, ylab="f(x)", main="Uniform(0,1)")
dunif(1)
dunif(0)
dunif(0.5)
dunif(2)
dunif(-1)
```

##### The Uniform (0,1) cumulative distribution function

```{r}
curve(punif(x), from=-1, to=2, ylab="P(X < x)", main="CDF")
punif(1)
punif(0)
punif(0.5)
```

##### The Uniform (0,1) inverse cumulative distribution function

```{r}
curve(qunif(x), main="Inverse CDF")
qunif(0.5)
qunif(0.95)
qunif(0.05)
```

##### The Uniform (0,360)

```{r}
curve(dunif(x,0,360), from=-40, to=400, ylab="f(x)", main="Uniform(0,360)")
1/360
curve(punif(x,0,360), from=-40, to=400, ylab="P(X < x)", main="CDF")
curve(qunif(x,0,360), main="Inverse CDF")
qunif(0.5,0,360)
```