How to create a probability density function and expectation in r?

Question

Task:

Eric has a friend, Ernie. Suppose that two flies sit in independent places, evenly distributed on the surface of the globes. Let D denote the Euclidean distance between Eric and Ernie (i.e., On a straight line along the inside of the globe).

Make a hypothesis about the probability density function D and give an estimate of its expected value, E (D).

So far I have created a function to create two points on the surface of the globe, but I'm not sure what to do next:

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)

Answer 1

This is an interesting problem. I will describe a computational approach; I will leave the math up to you.

• First we fix a random seed for reproducibility.

  set.seed(2018);
  

• 10^4 .

  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);
  

  , sample3d , .

• 2 df, ( dist) N = 10^4 .

  # Sample 2 points randomly from df, repeat N times
  N <- 10^4;
  dist <- replicate(N, dist(df[sample(1:nrow(df), 2), ]));
  

  @JosephWood, N = 10^4 . bootstrap . N -> infinity , , () ( Bootstrap). 1/sqrt(N), N = 10^4 1%.

• :

  # Let plot the distribution
  ggplot(data.frame(x = dist), aes(x)) + geom_histogram(bins = 50);
  

4. , .

   # Mean
   mean(dist);
   #[1] 1.333021
   
   # Median
   median(dist);
   #[1] 1.41602
   

   :

   mean.th = 4/3
   median.th = sqrt(2)