# # Distribution Functions

## # Normal distribution

Let's use `*norm`

as an example. From the documentation:

```
dnorm(x, mean = 0, sd = 1, log = FALSE)
pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
rnorm(n, mean = 0, sd = 1)
```

So if I wanted to know the value of a standard normal distribution at 0, I would do

```
dnorm(0)
```

Which gives us `0.3989423`

, a reasonable answer.

In the same way `pnorm(0)`

gives `.5`

. Again, this makes sense, because half of the distribution is to the left of 0.

`qnorm`

will essentially do the opposite of `pnorm`

. `qnorm(.5)`

gives `0`

.

Finally, there's the `rnorm`

function:

```
rnorm(10)
```

Will generate 10 samples from standard normal.

If you want to change the parameters of a given distribution, simply change them like so

`rnorm(10, mean=4, sd= 3)`

## # Binomial Distribution

We now illustrate the functions `dbinom`

,`pbinom`

,`qbinom`

and `rbinom`

defined for **Binomial distribution**.

The `dbinom()`

function gives the probabilities for various values of the binomial variable. Minimally it requires three arguments. The first argument for this function must be a vector of quantiles(the possible values of the random variable `X`

). The second and third arguments are the `defining parameters`

of the distribution, namely, `n`

(the number of independent trials) and `p`

(the probability of success in each trial). For example, for a binomial distribution with `n = 5`

, `p = 0.5`

, the possible values for X are `0,1,2,3,4,5`

. That is, the `dbinom(x,n,p)`

function gives the probability values `P( X = x )`

for `x = 0, 1, 2, 3, 4, 5`

.

```
#Binom(n = 5, p = 0.5) probabilities
> n <- 5; p<- 0.5; x <- 0:n
> dbinom(x,n,p)
[1] 0.03125 0.15625 0.31250 0.31250 0.15625 0.03125
#To verify the total probability is 1
> sum(dbinom(x,n,p))
[1] 1
>
```

The binomial probability distribution plot can be displayed as in the following figure:

```
> x <- 0:12
> prob <- dbinom(x,12,.5)
> barplot(prob,col = "red",ylim = c(0,.2),names.arg=x,
main="Binomial Distribution\n(n=12,p=0.5)")
```

Note that the binomial distribution is symmetric when `p = 0.5`

. To demonstrate that the binomial distribution is negatively skewed when `p`

is larger than `0.5`

, consider the following example:

```
> n=9; p=.7; x=0:n; prob=dbinom(x,n,p);
> barplot(prob,names.arg = x,main="Binomial Distribution\n(n=9, p=0.7)",col="lightblue")
```

When `p`

is smaller than `0.5`

the binomial distribution is positively skewed as shown below.

```
> n=9; p=.3; x=0:n; prob=dbinom(x,n,p);
> barplot(prob,names.arg = x,main="Binomial Distribution\n(n=9, p=0.3)",col="cyan")
```

We will now illustrate the usage of the cumulative distribution function `pbinom()`

. This function can be used to calculate probabilities such as `P( X <= x )`

. The first argument to this function is a vector of quantiles(values of x).

```
# Calculating Probabilities
# P(X <= 2) in a Bin(n=5,p=0.5) distribution
> pbinom(2,5,0.5)
[1] 0.5
```

The above probability can also be obtained as follows:

```
# P(X <= 2) = P(X=0) + P(X=1) + P(X=2)
> sum(dbinom(0:2,5,0.5))
[1] 0.5
```

To compute, probabilities of the type: `P( a <= X <= b )`

```
# P(3<= X <= 5) = P(X=3) + P(X=4) + P(X=5) in a Bin(n=9,p=0.6) dist
> sum(dbinom(c(3,4,5),9,0.6))
[1] 0.4923556
>
```

Presenting the binomial distribution in the form of a table:

```
> n = 10; p = 0.4; x = 0:n;
> prob = dbinom(x,n,p)
> cdf = pbinom(x,n,p)
> distTable = cbind(x,prob,cdf)
> distTable
x prob cdf
[1,] 0 0.0060466176 0.006046618
[2,] 1 0.0403107840 0.046357402
[3,] 2 0.1209323520 0.167289754
[4,] 3 0.2149908480 0.382280602
[5,] 4 0.2508226560 0.633103258
[6,] 5 0.2006581248 0.833761382
[7,] 6 0.1114767360 0.945238118
[8,] 7 0.0424673280 0.987705446
[9,] 8 0.0106168320 0.998322278
[10,] 9 0.0015728640 0.999895142
[11,] 10 0.0001048576 1.000000000
>
```

The `rbinom()`

is used to generate random samples of specified sizes with a given parameter values.

```
# Simulation
> xVal<-names(table(rbinom(1000,8,.5)))
> barplot(as.vector(table(rbinom(1000,8,.5))),names.arg =xVal,
main="Simulated Binomial Distribution\n (n=8,p=0.5)")
```

#### # Remarks

There are generally four prefixes:

**d**-The**density**function for the given distribution**p**-The cumulative distribution function**q**-Get the**quantile**associated with the given probability**r**-Get a**random**sample

For the distributions built into R's base installation, see `?Distributions`

.

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