硬币翻转的3d直方图产生R

Question

My goal is to generate a 3d histogram of the probability of obtaining a certain number of heads in a first and second sequence of 4 coin flips each.

My idea is simple:

• use expand.grid to take the cartesian product of the probabilities of a certain number of heads in the first and second sequence

• apply a product operation to each item in the cartesian product to obtain the probability of having that many heads in the first AND that many heads in the second. I think the trouble is in this step.

• show them in a 3d histogram.

But I get an extremely confusing output I cannot comprehend:

I would expect and output with bars lower at the extremes (hard to get 0 head in the first AND in the second) and high in the middle (easier to get 2 heads in the first AND in the second).

require("plot3D")

x <- c(1, 4, 6, 4, 1)/8
y <- c(1, 4, 6, 4, 1)/8

prod <- function( arr ) { return (arr[1]*arr[1])}

z <- as.matrix(apply(expand.grid(x,y), c(1,2), prod))

print(z)
##  Plot as a 3D histogram:
hist3D(z=z, border="black")

Also the output of expand.grid looks too linear, instead of a table as a cartesian product should:

    Var1  Var2
1  0.125 0.125
2  0.375 0.125
3  0.375 0.125
4  0.125 0.125
5  0.125 0.375
6  0.375 0.375
7  0.375 0.375
8  0.125 0.375
9  0.125 0.375
10 0.375 0.375
11 0.375 0.375
12 0.125 0.375
13 0.125 0.125
14 0.375 0.125
15 0.375 0.125
16 0.125 0.125

Answer 1

Given x and y are the number of heads in 1st and 2nd sequence of coin flips respectively coming from independent binomial distributions .

You need to evaluate z as a joint probability distribution over the grid of x and y , where x, y € {0,1,2,3,4}

require("plot3D")

x_val <- 0:4                    # No of heads in 1st sequence of flips
y_val <- 0:4                    # No of heads in 2nd sequence of flips

px <- c(1, 4, 6, 4, 1)/16       # probability vector for no of flips 
py <- c(1, 4, 6, 4, 1)/16       # in 1st and 2nd sequence of flips

grid_prob <- mesh(px, py)
z  <- with(grid_prob, x*y)      # sum(z) should equal 1 to be a pdf

# Plot as a 3D histogram:
hist3D(z=z, border="black", x = x_val , y = y_val)