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The probability density function I have is as following:

enter image description here

Now, I want to rewrite the cumulative probability function bellow into R.

enter image description here

The variables ($S$, $\mu$, $\sigma$) are constant variables. For $t$ i.e. t <- seq(0.1,1,0.1)

How the cumulative probability would look like in R? Non of my solutions yields desired output!


f1 = function(t,S,sigma,mu){((mu*t+S)/(sigma*sqrt(t)))}
f2 = function(t,S,sigma,mu){((mu*t-S)/(sigma*sqrt(t)))}


This kind of gives me reasonable results but not sure if correct.

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This is not a CDF as it does not converge to 1 if t goes to infinity. –  Michael Mayer Jan 12 at 19:10
You are right! What I need is cumulative probability and not CDF! –  Max Jan 12 at 19:11
Max - (i) what do you mean by cumulative probability if not CDF? (ii) Where do your equations come from? Are you sure you have them right? –  Glen_b Jan 12 at 21:00
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2 Answers

up vote 1 down vote accepted

This would implement your function in R. What part of the output didn't make sense when you tried?

 cumulative = function(t){
      S = somenumber
      mu = somenumber
      sigma = somenumber
      cd = pnorm((mu*t+S)/(sigma*sqrt(t)))-exp(-2*S*mu/sigma^2)*pnorm((mu*t-S)/(sigma*sqrt(t)))

That being said, what are your parameter values? For combinations I've tried, this seems like a decreasing function of t.

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I'm just trying to replicate someone elses work but don't have the complete workouts so I'm second guessing some things here. –  Max Jan 12 at 20:03
My solution (see EDIT) provides me with the desired outcome (same as yours just with the 1-(pnorm...) so something must be missing here. In general your answer is correct. Thanks. –  Max Jan 12 at 20:11
@Max Your problem is in the pnorm call. You call pnorm(f1(...),mu,sigma,0,1) so you're not computing from a N(0,1) distribution, but from a N(mu,sigma) distribution. Also by including the 1,0 in your call, you are reversing the calculation, pnorm(2,mu,sigma,0,0)=1-pnorm(2,mu,sigma,1,0) –  ltronneberg Jan 12 at 20:45
@Max And we dont get the same results, atleast not when I tried to compare the two. –  ltronneberg Jan 12 at 20:55
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A CDF is defined as non-decreasing, right continuous function F, with

  • $\lim_{x\to -\infty}F(x)=0,$ and

  • $\lim_{x\to +\infty}F(x)=1.$

I am just stating the obvious here, but when we let $t<0$ we have already a problem as your real-valued function contains square roots of t. Similarly, we need $\sigma >0$ and perhaps a lot of other things.

In either case, I will just go ahead and assume you know what you are doing, and show you the code you asked for.

# custom cdf
custom.cdf <- function(S,sigma,mu,t)
    q1 <- pnorm((mu*t + S)/(sigma*sqrt(t)))
    q2 <- pnorm((mu*t - S)/(sigma*sqrt(t)))
    e.term <- exp(-((2*S*mu)/(sigma^2)))
    res <- q1 - e.term*q2

# set parameters
S <- 1
sigma <- 1
mu <- 2    

t <- seq(0.1,1,0.1)

I have chosen a larger support to see a bit more of the curve. Check the formula (mine and yours) for any mistakes.

t <- seq(1e-6,1,1e-6)
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Shouldn't q1 <- pnorm((mu*t + S)/sigma*sqrt(t)) be pnorm((mu*t+S)/(sigma*sqrt(t)))? As sqrt(t) is also in the denominator –  ltronneberg Jan 12 at 20:03
Yes, spot on. I was getting even more spectacular outputs here :) –  Max Jan 12 at 20:05
Yes that's correct. Thanks! I will edit and get rid of the graphic. –  means-to-meaning Jan 12 at 20:16
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