**Calculations under Distributions**

**Leslie Hendrix**

August, 2020

August, 2020

# Calculations under Distributions

The ‘d’ functions in R calculate the the PMF or PDF values for a distribution.

- f(x) = P(X = x) for discrete distributions, but NOT a probability for continuous distributions

The ‘p’ functions in R calculate the CDF

- F(x) = P(X <= x)

The ‘q’ functions in R calcuate a percentile, that is they take the area to the left, and return a value in the distribution.

## Binomial

dbinom(j,n,p) gives P(Y = j) and pbinom(J,n,p) gives P(<=J) = P(Y = 0) + P(Y = 1) + … + P(Y = J)

## Poisson

dpois(j,lambda) gives P(Y=j) and

ppois(J,lambda) gives P(Y<=J)

## Continuous Uniform

punif(x,a,b) gives P(X<=x) in the uniform distribution with lower limit, a, and upper limit b

qunif(p,a,b) gives the pth percentile

## Exponential

pexp(x,lambda) gives P(Y<=j)

qexp(p, lambda) gives the pth percentile

## Normal

pnorm(x,mu,sigma) gives Pr{X < x} for X~N(mu,sigma) …..so, 1-pnorm(x,mu,sigma) gives Pr(X > x)

qnorm(p,mu,sigma) gives the value in the normal distribution (with mean mu and sd sigma) that has p to the left of it

## t distribution

pt(t,df) gives Pr{T < t} for T~t(df)

qt(p,df) gives the value of the t distribution with df=df that has p to the left of it

# Tips

- Note that the ‘d’ functions calculate the height of the curve at that X value for a continuous distribution. This is NOT a probability.
- The ‘q’ and ‘p’ functions do the opposite calculation.
- The ‘p’ functions take a value in a distribution and return the area to the left.
- The ‘q’ functions take the area to the left and return a value in the distribution.