HW5

library(matlib)
library(MASS)
require(plotrix)
require(grid)
library(ggplot2)
require(ggthemes)

5a

B+C < 1/2

B + C < 1/2 passes through the points (1/2, 0) and (0, 1/2)

Solving by geometry:

The probability is therefore the area under the line of b= 1/2 - C

ans:

curve(0.5-x, from=0.0, to=0.5, , xlab="B", ylab="C",main = "B+C < 1/2")

5b

B*C < 1/2

Using integration for the curve:

But to also capture the area to the left of the curve, which is 1/2*1 = 0.5

Therefore the Total area is: 0.5 + area under the curve (from the integral above):

ans:

func2 <- function(x){
  1/(2*x)
}

integrate(func2, 1/2,1)
## 0.3465736 with absolute error < 3.8e-15
#The area under the hyperbola curve is graphed below
curve(1/(2*x) , from=0.5, to=1, , xlab="B", ylab="C",main = "B < 1/(2*C)")

#The total area of the integral and 1/2 of the square 
blue<-rgb(0.8, 0.8, 1, alpha=0.25)
clear<-rgb(1, 0, 0, alpha=0.0001)

func1<-function(x) 0.5/x
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x,fill='clear')) + 
 stat_function(fun = func1) + xlim(0,1) + ylim(0,1)+ggtitle("B*C < 1/2")+ theme_economist()+

  geom_label(aes(x = .7, y = .85, label = "Subtracted area"), 
             hjust = 0, 
             vjust = 0.5, 
             colour = "blue", 
             fill = "white", 
             label.size = NA, 
             family="Helvetica", 
             size = 6)

L1 <- geom_segment(aes(x=0,xend=0,y=0,yend=1))
L2 <- geom_segment(aes(x=0,xend=0.5,y=1,yend=1))
L3 <- geom_segment(aes(x=0,xend=1,y=0,yend=0))
L4 <- geom_segment(aes(x=1,xend=1,y=0,yend=0.5))

p + L1 + L2 + L3 + L4
## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

5c

|B − C| < 1/2

ans:

|B-C| < 1/2 is the region between the lines C = B+1/2 and C = B-1/2. These 2 regions intersect the unit square

Note that this is equivalent to B - 1/2 < C < x + 1/2.

So, we need to subtract these 2 triangles from the unit square => 1 - 2 * (1/2 * 1/2 * 1/2) = 3/4

func<-function(x) x-0.5
func2<-function(x) x+0.5
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x)) + 
 stat_function(fun = func) + xlim(0,1) + ylim(0,1)+stat_function(fun = func2)+ggtitle("|B-C| < 1/2")+ theme_economist()+
  geom_label(aes(x = 0.01, y = .86, label = "1st subtracted triangle"), 
             hjust = 0, 
             vjust = 0.5, 
             colour = "blue", 
             fill = "white", 
             label.size = NA, 
             family="Helvetica", 
             size = 4)+
  geom_label(aes(x = 0.67, y = 0.125, label = "2nd subtracted triangle"), 
             hjust = 0, 
             vjust = 0.5, 
             colour = "blue", 
             fill = "white", 
             label.size = NA, 
             family="Helvetica", 
             size = 4)

L1 <- geom_segment(aes(x=0,xend=0,y=0,yend=1))
L2 <- geom_segment(aes(x=0,xend=1,y=1,yend=1))
L3 <- geom_segment(aes(x=0,xend=1,y=0,yend=0))
L4 <- geom_segment(aes(x=1,xend=1,y=0,yend=1))


p+ L1 + L2 + L3 + L4
## Warning: Removed 50 rows containing missing values (geom_path).

## Warning: Removed 50 rows containing missing values (geom_path).
## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

5d

P(max(B,C) <1/2)

Is like saying max{B,C} < 1/2 is equivalent to: B < 1/2 and C < 1/2

P(B<=1/2)*P(C<=1/2)

The area of Two squares (both red shaded and green shaded, see below) with sides of 1/2 is:

=> 1/2 * 1/2 => 1/4

plot(c(0, 2), c(0, 2), type= "n", xlab = "B", ylab = "C",main = "max{B,C} < 1/2",lwd = 15)
rect(0, 0, 0.5,0.5 , density = 17, border = "darkred",lty =2, col='red')
rect(0,0,0.5,.5,density = 12, border = "green",lty =4, col = 'green')
rect(0, 0, 1,1 , density = 0, border = "blue")

5e

Again, Solving by geometry:

P(min(B,C) <1/2)

Is like saying B < 1/2 or C < 1/2

The area of Two squares with the red shaded one being a 1x1/2 and the green shaded on being 1/2x1/2:

=> 1/2 + 1/4 => 3/4

plot(c(0, 2), c(0, 2), type= "n", xlab = "B", ylab = "C",main = "min{B,C} < 1/2",lwd = 10)
rect(0, 0, .5,1 , density = 10, border = "red",lty =4, col ="red")
rect(0.5,0,1,.5,density = 7, border = "darkgreen",lty =4,col="green")
rect(0, 0, 1,1 , density = 0, border = "blue")