How to plot the probabilistic density function of a function?

Question

Assume A follows Exponential distribution; B follows Gamma distribution How to plot the PDF of 0.5*(A+B)

Answer 1

I'm not an R programmer, but it might be helpful to know that for independent random variables with PDFs f₁(x) and f₂(x), the PDF of the sum of the two variables is given by the convolution f₁ * f₂ (x) of the two input PDFs.

Answer 2

Here is an attempt at doing the convolution (which @Jim Lewis refers to) in R. Note that there are probably much more efficient ways of doing this.

lower <- 0
upper <- 20
t <- seq(lower,upper,0.01)
fA <- dexp(t, rate = 0.4)
fB <- dgamma(t,shape = 8, rate = 2)
## C has the same distribution as (A + B)/2
dC <- function(x, lower, upper, exp.rate, gamma.rate, gamma.shape){
  integrand <- function(Y, X, exp.rate, gamma.rate, gamma.shape){
    dexp(Y, rate = exp.rate)*dgamma(2*X-Y, rate = gamma.rate, shape = gamma.shape)*2
  }
  out <- NULL
  for(ix in seq_along(x)){
    out[ix] <-
      integrate(integrand, lower = lower, upper = upper,
                X = x[ix], exp.rate = exp.rate,
                gamma.rate = gamma.rate, gamma.shape = gamma.shape)$value
  }
  return(out)
}
fC <- dC(t, lower=lower, upper=upper, exp.rate=0.4, gamma.rate=2, gamma.shape=8)
## plot the resulting distribution
plot(t,fA,
     ylim = range(fA,fB,na.rm=TRUE,finite = TRUE),
     xlab = 'x',ylab = 'f(x)',type = 'l')
lines(t,fB,lty = 2)
lines(t,fC,lty = 3)
legend('topright', c('A ~ exp(0.4)','B ~ gamma(8,2)', 'C ~ (A+B)/2'),lty = 1:3)

Answer 3

If you're just looking for fast graph I usually do the quick and dirty simulation approach. I do some draws, slam a Gaussian density on the draws and plot that bad boy:

numDraws   <- 1e6
gammaDraws <- rgamma(numDraws, 2)
expDraws   <- rexp(numDraws)
combined   <- .5 * (gammaDraws + expDraws)
plot(density(combined))

output should look a little like this: