Computational Methods for Finding Probabilities of Intervals

Transcription

1 Computational Methods for Finding Probabilities of Intervals Often the standard normal density function is denoted ϕ(z) = (2π) 1/2 exp( z 2 /2) and its CDF by Φ(z). Because Φ cannot be expressed in closed form, areas under the normal curve corresponding to probabilities of the form P{a < Z b} are found by numerical integration, simulation or rational approximation. Here we use R to illustrate some of these methods, evaluating J = P{0 < Z 1} as an example, where Z ~ NORM(0, 1). A slight modification of the first method below is used in the pnorm function of R. This function can be used to evaluate J as pnorm(1) - pnorm(0), which returns

2 Depending on their format, standard printed tables can be used to get P{0 < Z < 1} = directly or to find it after a simple computation, such as = For many practical applications, accuracy beyond 2 or 3 decimal places is not required. The methods illustrated here for the standard normal distribution can also be used for other continuous distributions where direct integration is not possible. Note: In R the CDFs of some other families of distributions are called punif, pbeta, pexp, pgamma, and pweibull. Use?punif, and so on, for an explanation of syntax, default parameters, and specification of parameters.

3 Numerical integration. By selecting a grid of a large number n of evenly spaced points z i in (a, b], one may approximate J by J Σ i w i h i, where w i = (b a)/n is the width of the base of a small rectangle containing z i and h i = ϕ(z i ) is its height. n <- 100; a <- 0; b <- 1; w <- (b - a)/n z <- seq(a + w/2, b - w/2, length=n) h <- dnorm(z) j <- sum(w*h); j [1] Using n = 1000 grid points, we obtain [Note that dnorm(z) could be replaced by exp(-z^2/2)/sqrt(2*pi).] This is a deterministic computation; every run with the same n will give the same result. Exercise 1: Use numerical integration to evaluate P{0 < Z < 1/2}.

4 Monte Carlo integration. Sometimes, especially in the approximation of integrals over high dimensional regions, it is easier and more accurate to select the z i at random on (a, b], rather than to use an evenly spaced grid. In one dimension, as here, this method tends to be less efficient than numerical integration and so we use a larger n = n < ; a <- 0; b <- 1; w <- (b - a)/n z <- runif(n, a, b); h <- dnorm(z) j <- sum(w*h); j [1] Because this is a simulation, each result will be a little different. Here it is reasonable to expect 2 or 3 place accuracy on each run. Other runs gave: , , , Exercise 2: (a) Use Monte Carlo integration to evaluate P{0 < Z < 1/2}. (b) Let X have the density f(x) = 6x(1 x), for 0 x 1. Use calculus, numerical integration, and Monte Carlo integration to evaluate or approximate P{X < 1/3}. What are your options if f(x) = x 0.9 (1 x) 1.1, for 0 x 1?

5 Acceptance-Rejection method. The desired area J above (0, 1] and beneath ϕ is wholly contained in the rectangle with base (0, 1) and height 0.4. We can generate many points randomly distributed in this interval, find the proportion of them in the desired area (the "accepted" points) and multiply that proportion by 0.4. With n = , this method gives 2 or 3 place accuracy for our example. n < ; y <- runif(n, 0,.4); z <- runif(n, 0, 1) j <- mean(y < dnorm(z)) * 0.4; j [1] Other runs gave: , , ,

6 A similar method can be used to get an approximation to π. The first quadrant of the unit circle, which has area π/4, is entirely contained in a square that has area 1. So π can be approximated if we multiply by 4 the fraction of random points in the square that falls in the first quadrant of the circle. n < x <- runif(n, 0, 1); y <- runif(n, 0, 1) pie <- mean(y < sqrt(1 - x^2)) * 4; pie [1] Other runs gave: , , Exercise 3: The Taylor expansion of e x is 1 x + x 2 /2! x 3 /3! + x 4 /4! Show that the area above the curve f(x) = 1 x + x 2 /2! x 3 /3! + x 4 /4! x 5 /5! + x 6 /6! within the unit square [vertices at (0, 0), (0, 1), (1, 0), and (1,1)] is approximately 1/e. Use this fact and the acceptance-rejection method to approximate 1/e thence e. Where is the greater error: (i) in using f(x) for e x or (ii) in the random sampling of points? Explain.

7 Sampling method. We can also simulate a sample from NORM(0, 1) and take the proportion lying in the desired interval (0, 1]. n < ; z <- rnorm(n, 0, 1) j <- mean(z > 0 & z < 1); j [1] Even with larger n, we can expect only about 2 place accuracy in our example. Other runs gave: , , , This method is useful in cases where we do not know the density function of the distribution under consideration, but can find a way to simulate it. For example, it is tedious to find the density function of the random variable X = U + V + W, where U ~ UNIF(0,2), V ~ EXPM(1), and W ~ NORM(2, 1/2), are independent random variables. But it wouldn't be difficult to simulate a sample of several thousand from the distribution of X.

8 n < u <- runif(n, 0, 2) v <- rexp(n, 1) w <- rnorm(n, 2, 1/2) x <- u + v + w hist(x) mean(x < 2) mean(x) var(x) sqrt(var(x)) > mean(x < 2) [1] Exact answer:????? > mean(x) [1] Exact answer: 4 > var(x) [1] Exact answer: > sqrt(var(x)) [1] Exact answer: Exercise 4: Approximate P{X < 3}.

9 Rational approximation. This method involves expanding the integrand, for example into an infinite series, the first few terms of which are integrated to give an approximation. Stegun and Abramowitz (Handbook of Mathematical Functions, 8th printing, National Bureau of Standards, 1970, page 932) show a rational approximation, which they claim is good to 6 or 7 places. Coding this approximation into R, we get an approximation of P{0 < Z < b}: c1 < ; c2 < ; c3 < ; c4 < ; c5 < ; d < b <- 1; t <- 1/(1 + d*b) j <-.5 - dnorm(b)*(c1*t + c2*t^2 + c3*t^3 + c4*t^4 + c5*t^5) j [1] This is a deterministic approximation; every run gives the same answer. There are many other rational approximations for Φ, and also for other continuous CDFs that do not have closed form. Exercise 5: For z in [0,.5), why is it reasonable to approximate Φ(z) by (2π) 1/2 (z z 3 /6 + z 5 /40 z 7 /168 + z 9 /3456 z 11 /42528) + 1/2? With R or a hand calculator, use this approximation to find Φ(1/4). Copyright 2004 by Bruce E. Trumbo. All rights reserved. Department of Statistics, CSU Hayward.