[英]How to generate a probability density function and expectation in r?
The task: 任务:
Eric the fly has a friend, Ernie. Eric the fly有一个朋友,Ernie。 Assume that the two flies sit at independent locations, uniformly distributed on the globe's surface. 假设两只苍蝇坐在独立的位置,均匀分布在地球表面。 Let D denote the Euclidean distance between Eric and Ernie (ie, on a straight line through the interior of the globe). 设D表示Eric和Ernie之间的欧几里德距离(即,穿过地球内部的直线)。
Make a conjecture about the probability density function of D and give an estimate of its expected value, E(D). 对D的概率密度函数进行猜想,并给出其期望值E(D)的估计。
So far I have made a function to generate two points on the globe's surface, but I am unsure what to do next: 到目前为止,我已经完成了在地球表面生成两个点的功能,但我不确定下一步该做什么:
sample3d <- function(2)
{
df <- data.frame()
while(n > 0){
x <- runif(1,-1,1)
y <- runif(1,-1,1)
z <- runif(1,-1,1)
r <- x^2 + y^2 + z^2
if (r < 1){
u <- sqrt(x^2+y^2+z^2)
vector = data.frame(x = x/u,y = y/u, z = z/u)
df <- rbind(vector,df)
n = n- 1
}
}
df
}
E <- sample3d(2)
This is an interesting problem. 这是一个有趣的问题。 I'll outline a computational approach; 我将概述一种计算方法; I'll leave the math up to you. 我会把数学留给你。
First we fix a random seed for reproducibility. 首先,我们修复随机种子以获得可重复性。
set.seed(2018);
We sample 10^4
points from the unit sphere surface. 我们从单位球面采样10^4
个点。
sample3d <- function(n = 100) { df <- data.frame(); while(n > 0) { x <- runif(1,-1,1) y <- runif(1,-1,1) z <- runif(1,-1,1) r <- x^2 + y^2 + z^2 if (r < 1) { u <- sqrt(x^2 + y^2 + z^2) vector = data.frame(x = x/u,y = y/u, z = z/u) df <- rbind(vector,df) n = n- 1 } } df } df <- sample3d(10^4);
Note that sample3d
is not very efficient, but that's a different issue. 请注意, sample3d
效率不高,但这是一个不同的问题。
We now randomly sample 2 points from df
, calculate the Euclidean distance between those two points (using dist
), and repeat this procedure N = 10^4
times. 我们现在从df
随机抽取2个点,计算这两个点之间的欧几里德距离(使用dist
),并重复此过程N = 10^4
次。
# Sample 2 points randomly from df, repeat N times N <- 10^4; dist <- replicate(N, dist(df[sample(1:nrow(df), 2), ]));
As pointed out by @JosephWood, the number N = 10^4
is somewhat arbitrary. 正如@JosephWood指出的那样, N = 10^4
的数字有些随意。 We are using a bootstrap to derive the empirical distribution. 我们使用bootstrap来推导经验分布。 For N -> infinity
one can show that the empirical bootstrap distribution is the same as the (unknown) population distribution (Bootstrap theorem). 对于N -> infinity
可以证明经验自助分布与(未知)种群分布(Bootstrap定理)相同。 The error term between empirical and population distribution is of the order 1/sqrt(N)
, so N = 10^4
should lead to an error around 1%. 经验和人口分布之间的误差项为1/sqrt(N)
,因此N = 10^4
应导致1%左右的误差。
We can plot the resulting probability distribution as a histogram: 我们可以将得到的概率分布绘制为直方图:
# Let's plot the distribution ggplot(data.frame(x = dist), aes(x)) + geom_histogram(bins = 50);
Finally, we can get empirical estimates for the mean and median. 最后,我们可以得到平均值和中位数的经验估计值。
# Mean mean(dist); #[1] 1.333021 # Median median(dist); #[1] 1.41602
These values are close to the theoretical values: 这些值接近理论值:
mean.th = 4/3 median.th = sqrt(2)
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