Quantile function for binomial distribution?

Question

A test will succeed with a certain percentage. Now this test is repeated X number of times. I want to be able to get an estimate of the total number of succeeded test.

Given that I know both the probability of success and the X number of attempts, is it possible to symbolically compute this estimate?

I want to use this in a piece of software. Where I generate a random value [0, 1] and then use the requested formula as a mapping function.

From what I gathered I need a quantile function of binomial distribution? Can I compute or estimate that one symbolically?

Answer 1

Suppose a fair die is rolled $n = 60$ times, then the number $X$ of Ones seen is $X \sim \mathsf{Binom}(n=60, p = 1/6).$ The expected number of Ones seen is $E(X) = np = 60(1/6) = 10.$ The probability of seeing exactly 10 Ones is $$P(X = 10) = {60 \choose 10}\left(\frac 1 6\right)^{10} \left(\frac 5 6\right)^{50} = 0.1270131.$$

PDF: The binomial PDF (or PMF) is defined as $f_X(k) = {n \choose k}p^k(1-p)^{n-k},$ for $k = 0, 1, \dots, n.$ In R statistical software, $f_X(10),$ for $n=60,\, p=1/6,$ can be computed by using the displayed formula or by using the built-in PDF dbinom:

choose(60,10)*(1/6)^10*(5/6)^50
[1] 0.1370131
dbinom(10, 60, 1/6)
[1] 0.1370131

CDF: The CDF is defined as $F_X(k) = P(X \le k),$ for any real value of $k.$ In our example for $k < 0,$ we have $F_X(k) = 0$ and for $k > n = 60,$ we have $F_X(k) = 1.$ From $k=0$ through $60,$ the CDF (denoted pbinom in R) is a 'pure jump' function, which can be plotted as shown below:

ml="CDF of BINOM(60, 1/6)"
curve(pbinom(x, 60, 1/6), -.5, 60.5, lwd=2, ylab="CDF", xlab="k", main=ml, n=10001)
abline(h=0:1, col="green3");  abline(v=0, col="green3")
k = 0:60; cdf=pbinom(k, 60, 1/6); points(k, cdf, pch=20, col="blue")

Blue dots emphasize that the value of the function $F_X(k)$ is the upper end of each vertical segment.

Inverse CDF: The 'quantile function' of $\mathsf{Binom}(60, 1/6)$ is the inverse of the CDF. In general, $F_X^{-1}(q) = c$ means $P(X \le c) = F_X(c) = q,$ for $0 < q < 1.$ In our example, the quantile function of $X$ can be used to get an interval in which values of $\mathsf{Binom}(60, 1/6)$ will lie with probability (just barely over) 95%. Specifically, $P(5 \le X \le 16) = P(X \le 16) - P(X \le 4) = 0.96.$ (Because of the discreteness of the binomial distribution it is not possible to get probability 0.95 exactly.)

qbinom(c(.025,.975), 60, 1/6) 
[1]  5 16
diff(pbinom(c(4,16), 60, 1/6))
[1] 0.9633622

The plot of the quantile function is shown below with horizontal dotted lines at 5 and 16:

ml = "Quantile function of BINOM(60, 1/6)"
curve(qbinom(x, 60, 1/6), 0, 1, lwd=2, n=10001, ylab="k", xlab="quantile", main=ml)
abline(v=0:1, col="green2");  abline(h = 0, col="green2")

Using the quantile function to simulate a random sample: Finally, you can use the quantile function to simulate values of $X \sim \mathsf{Binom}(60, 1/6)$ from values of $U \sim \mathsf{Unif}(0,1).$ Below we use 10,000 standard uniform random variables to simulate as many values of $X,$ for which $\bar X \approx E(X) = 10.$ (Notice that values of $X$ above about 20 or 25 are very rare.)

set.seed(518);  m = 10000;  u = runif(m);  x = qbinom(u, 60, 1/6;  mean(x)
[1] 9.9993  # aprx E(X) = 10
summary(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.000   8.000  10.000   9.999  12.000  23.000 
quantile(x, .999)
99.9% 
   20 

Below is a histogram of the simulated random sample of 10,000 observations from $\mathsf{Binom}(60, 1/6);$ centers of superimposed open circles show exact binomial probabilities. (The fit is about as good as can be anticipated for a sample of 10,000; a sample of a million values would give a much better match.)

ml="Random Sample of 10,000 Observations from BINOM(60, 1/6)"
hist(x, prob=T, br=(min(x):(max(x)+1))-.5, col="skyblue2", main=ml)
  k=0:25; pdf = dbinom(k, 60, 1/6)
  points(k, pdf, col="red")

Notes: (a) If you use the same seed (in set.seed) as mine for the pseudorandom number generator in R, you will get exactly the same 10,000 observations I did. If you set a different seed, your results may be a little closer to (or farther from) a perfect fit than mine. [If you set no seed, R will pick an undisclosed, unpredictable seed from the system clock.] (b) R has a built-in function rbinom for generating random binomial samples. Essentially, runif(qbinom(m, n, p)) is the same as rbinom(m, n, p), but the latter has slight modifications for more efficient running.