Probability of completing a task along certain path in a certain time using beta distribution I have 5 different tasks and the duration of each task in the network are assumed to be beta distributed and I have their beta parameters (p, q, a, b) for each task. These durations are assumed to be independent of each other.
How do I find the probability of completing the first 3 tasks in a certain time?
Any help would be appreciated!
 A: If you add a lot of beta distributed variables, the sum will eventually follow a normal distribution. But tree will not be enough to go that route.
If the parameters of the beta distributions are given, then it is very easy to set up a computer simulation of the sum of three such distributions.
In R that would be:
# let a to f be the parameters of the beta distributions be know. We assume some random values
a <- 5; b <- 3; c <- 10; d <- 2; e <- 5; f <- 8
n <- 100000

# we draw n triples of beta values and sum them
sum_of_betas <- rbeta(n, a, b) + rbeta(n, c, d) + rbeta(n, e, f)

# Plot histogram of the resulting sums:
hist(sum_of_betas)

# To compute the share of sums that are below 1.5 we can then do
sum(sum_of_betas < 1.5)/n

This is simulation, not a closed form solution. The advantage of this is, that it can easily be adjusted to more complicated situations, if needed.
Edit
Answering the comment: The simulation is based on $n$ replicates. So if you look for small probabilities, you might want to increase $n$ so large, that the simulation output is very similar each time you run the simulation.
In addition to what I coded above, you want beta distributions that do not only range from 0 to 1 but have two more parameters, giving the minimum and the maximum of the distribution. I therefore added variables named minA and minBand so on for the additional parameters. The code then becomes:
# let a to f be the shape parameters of the beta distributions be know. We assume some random values
a <- 5; b <- 3; c <- 10; d <- 2; e <- 5; f <- 8
# let minA to min C be the minimum and maxA to maxC be the maximal values of each beta distr.
minA <- 0; minB <- 1; minC <- 3; maxA <- 1; maxB <- 2; maxC <- 6
# number of random triples is n
n <- 1000000

# we draw n triples of beta values and sum them
sum_of_betas <- minA + (maxA-minA)*rbeta(n, a, b)+
                minB + (maxB-minB)*rbeta(n, c, d) + 
                minC + (maxC-minC)*rbeta(n, e, f)

# Plot histogram of the resulting sums:
hist(sum_of_betas)

# To compute the share of sums that are below 7 we can then do
sum(sum_of_betas < 7)/n

