Homework 2

NAME: Alexander Jenson

EXERCISE 1

x <- c(rep(0.01,26), rep(0.01,6), rep(0.021,2), rep(0.01,10), 0.022, rep(0.041,2), rep(0.024,2), rep(0.01,7), 0.014, 0.043, 0.08, 0.082, rep(0.038,2), 0.027, rep(0.01,6), 0.025, 0.037, 0.11, 0.18, 0.081, rep(0.039,2), 0.013, rep(0.01,5), 0.026, 0.036, 0.079, 0.09, rep(0.04,2), 0.016, rep(0.01,7), 0.015, 0.044, 0.078, 0.04, 0.018, rep(0.01,8), rep(0.019,2), rep(0.042,2), 0.016, rep(0.01,55))
priorMat <- matrix(x, nrow=13) / 3
priorMat

##              [,1]        [,2]        [,3]        [,4]        [,5]
##  [1,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [2,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [3,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [4,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [5,] 0.003333333 0.003333333 0.003333333 0.003333333 0.004666667
##  [6,] 0.003333333 0.003333333 0.003333333 0.007333333 0.014333333
##  [7,] 0.003333333 0.003333333 0.007000000 0.013666667 0.026666667
##  [8,] 0.003333333 0.003333333 0.007000000 0.013666667 0.027333333
##  [9,] 0.003333333 0.003333333 0.003333333 0.008000000 0.012666667
## [10,] 0.003333333 0.003333333 0.003333333 0.008000000 0.012666667
## [11,] 0.003333333 0.003333333 0.003333333 0.003333333 0.009000000
## [12,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
## [13,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##              [,6]        [,7]        [,8]        [,9]       [,10]
##  [1,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [2,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [3,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [4,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##  [5,] 0.008333333 0.008666667 0.003333333 0.003333333 0.003333333
##  [6,] 0.012333333 0.012000000 0.005000000 0.006333333 0.003333333
##  [7,] 0.036666667 0.026333333 0.014666667 0.006333333 0.003333333
##  [8,] 0.060000000 0.030000000 0.026000000 0.014000000 0.003333333
##  [9,] 0.027000000 0.013333333 0.013333333 0.014000000 0.003333333
## [10,] 0.013000000 0.013333333 0.006000000 0.005333333 0.003333333
## [11,] 0.013000000 0.005333333 0.003333333 0.003333333 0.003333333
## [12,] 0.004333333 0.003333333 0.003333333 0.003333333 0.003333333
## [13,] 0.003333333 0.003333333 0.003333333 0.003333333 0.003333333
##             [,11]       [,12]       [,13]
##  [1,] 0.003333333 0.003333333 0.003333333
##  [2,] 0.003333333 0.003333333 0.003333333
##  [3,] 0.003333333 0.003333333 0.003333333
##  [4,] 0.003333333 0.003333333 0.003333333
##  [5,] 0.003333333 0.003333333 0.003333333
##  [6,] 0.003333333 0.003333333 0.003333333
##  [7,] 0.003333333 0.003333333 0.003333333
##  [8,] 0.003333333 0.003333333 0.003333333
##  [9,] 0.003333333 0.003333333 0.003333333
## [10,] 0.003333333 0.003333333 0.003333333
## [11,] 0.003333333 0.003333333 0.003333333
## [12,] 0.003333333 0.003333333 0.003333333
## [13,] 0.003333333 0.003333333 0.003333333

sum(priorMat)

## [1] 1

oceanData <- data.frame(index=c(1:169),
            longitude=rep(c(1:13),each=13),
            latitude=rep(c(1:13),13),
            Day0prob=c(priorMat))

oceanData[10,]

##    index longitude latitude    Day0prob
## 10    10         1       10 0.003333333

oceanData[,2]

##   [1]  1  1  1  1  1  1  1  1  1  1  1  1  1  2  2  2  2  2  2  2  2  2  2
##  [24]  2  2  2  3  3  3  3  3  3  3  3  3  3  3  3  3  4  4  4  4  4  4  4
##  [47]  4  4  4  4  4  4  5  5  5  5  5  5  5  5  5  5  5  5  5  6  6  6  6
##  [70]  6  6  6  6  6  6  6  6  6  7  7  7  7  7  7  7  7  7  7  7  7  7  8
##  [93]  8  8  8  8  8  8  8  8  8  8  8  8  9  9  9  9  9  9  9  9  9  9  9
## [116]  9  9 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11
## [139] 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13
## [162] 13 13 13 13 13 13 13 13

oceanData[10,2]

## [1] 1

maxArea <- which.max(oceanData$Day0prob)
maxArea

## [1] 73

oceanData[maxArea,]

##    index longitude latitude Day0prob
## 73    73         6        8     0.06

suppressPackageStartupMessages(library(ggplot2))
ggplot(oceanData, aes(x = longitude, y = latitude)) + geom_raster(aes(fill = Day0prob)) + scale_fill_gradientn(colours=c("#4B0082", "#0000FF", "#00FF00", "#FFFF00", "#FF9900","#FF0000"))

EXERCISE 2

\(P(A_{73}) = 0.06\)

\(P(F_{73}|A_{73}) = \frac{1}{4}\)

\(P(F_{73}|A_{73}^c) = 1\)

\[\begin{split} P(F_{73}) & = P(F_{73}|A_{73})P(A_{73}) + P(F_{73}|A_{73}^c)P(A_{73}^c) \\ & = \frac{1}{4}*(0.06) + 1*(0.94) \\ & = 0.955 \end{split}\]

\[\begin{split} P(A_{73} | F_{73}) & = \frac{P(F_{73} | A_{73})P(A_{73})}{P(F_{73})} \\ & = \frac{(0.25)(0.06)}{(0.955)} & = 0.01570681 \end{split}\]

\[\begin{split} P(A_{74} | F_{73}) & = \frac{P(F_{73} | A_{74})P(A_{74})}{P(F_{73})} \\ & = \frac{(1)(0.027)}{0.955} \\ & = 0.028272 \end{split}\]
Area 73 was more likely than Area 74 a priori (0.06 v. 0.027), but after F73, Area 74 is more likely (0.028272 v. 0.014325)

EXERCISE 3

prior <- oceanData$Day0prob
area <- 73
day <- 1
prior[1]

## [1] 0.003333333

prior[73]

## [1] 0.06

probFail <- (day/(day+3))*prior[area] + (1-prior[area])

posterior = (1*prior)/probFail
posterior[area] = ((day/(day+3))*prior[area])/probFail

print(posterior)

##   [1] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##   [6] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [11] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [16] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [21] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [26] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [31] 0.003490401 0.003490401 0.007329843 0.007329843 0.003490401
##  [36] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [41] 0.003490401 0.003490401 0.003490401 0.003490401 0.007678883
##  [46] 0.014310646 0.014310646 0.008376963 0.008376963 0.003490401
##  [51] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [56] 0.003490401 0.004886562 0.015008726 0.027923211 0.028621291
##  [61] 0.013263525 0.013263525 0.009424084 0.003490401 0.003490401
##  [66] 0.003490401 0.003490401 0.003490401 0.003490401 0.008726003
##  [71] 0.012914485 0.038394415 0.015706806 0.028272251 0.013612565
##  [76] 0.013612565 0.004537522 0.003490401 0.003490401 0.003490401
##  [81] 0.003490401 0.003490401 0.009075044 0.012565445 0.027574171
##  [86] 0.031413613 0.013961606 0.013961606 0.005584642 0.003490401
##  [91] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
##  [96] 0.003490401 0.005235602 0.015357766 0.027225131 0.013961606
## [101] 0.006282723 0.003490401 0.003490401 0.003490401 0.003490401
## [106] 0.003490401 0.003490401 0.003490401 0.003490401 0.006631763
## [111] 0.006631763 0.014659686 0.014659686 0.005584642 0.003490401
## [116] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [121] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [126] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [131] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [136] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [141] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [146] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [151] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [156] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [161] 0.003490401 0.003490401 0.003490401 0.003490401 0.003490401
## [166] 0.003490401 0.003490401 0.003490401 0.003490401

sum(posterior)

## [1] 1

posterior[73]

## [1] 0.01570681

searchProb <- function(prior, area, day){
  probFail <- (day/(day+3))*prior[area] + (1-prior[area])
  posterior <- (1*prior)/probFail
  posterior[area] <- ((day/(day+3))*prior[area])/probFail
  return(posterior)
}

oceanData$Day1prob = searchProb(prior=oceanData$Day0prob, area=73, day=1)

ggplot(oceanData, aes(x=longitude, y=latitude)) +
    geom_raster(aes(fill = Day1prob)) + 
    scale_fill_gradientn(colours=c("#4B0082", "#0000FF", "#00FF00", "#FFFF00", "#FF9900","#FF0000"))

EXERCISE 4

next_ind = which.max(oceanData$Day1prob)

oceanData$Day2prob = searchProb(prior=oceanData$Day1prob, area = next_ind, day = 2)

ggplot(oceanData, aes(x=longitude, y=latitude)) +
    geom_raster(aes(fill = Day2prob)) + 
    scale_fill_gradientn(colours=c("#4B0082", "#0000FF", "#00FF00", "#FFFF00", "#FF9900","#FF0000"))

next_ind = which.max(oceanData$Day2prob)
oceanData$Day3prob = searchProb(prior=oceanData$Day2prob, area = next_ind, day = 3)
ggplot(oceanData, aes(x=longitude, y=latitude)) +
    geom_raster(aes(fill = Day3prob)) + 
    scale_fill_gradientn(colours=c("#4B0082", "#0000FF", "#00FF00", "#FFFF00", "#FF9900","#FF0000"))

next_ind = which.max(oceanData$Day3prob)
oceanData$Day4prob = searchProb(prior=oceanData$Day3prob, area = next_ind, day = 4)
ggplot(oceanData, aes(x=longitude, y=latitude)) +
    geom_raster(aes(fill = Day4prob)) + 
    scale_fill_gradientn(colours=c("#4B0082", "#0000FF", "#00FF00", "#FFFF00", "#FF9900","#FF0000"))

As each day passed, the posterior probability of finding the crash site in each of the areas NOT explored increased slightly, while the posteriors of areas that were searcehd shrunk considerably.

EXERCISE 5

\(x\)	1	2	3	4	5	6	Total
\(f_X(x)\)	\(\frac{11}{36}\)	\(\frac{9}{36}\)	\(\frac{7}{36}\)	\(\frac{5}{36}\)	\(\frac{3}{36}\)	\(\frac{1}{36}\)	\(\frac{36}{36}\)

\(y\)	1	2	3	4	5	6	Total
\(f_Y(y)\)	\(\frac{1}{36}\)	\(\frac{3}{36}\)	\(\frac{5}{36}\)	\(\frac{7}{36}\)	\(\frac{9}{36}\)	\(\frac{11}{36}\)	\(\frac{36}{36}\)

EXERCISE 6

\(f_{X,Y}(x,y)\)	1	2	3	4	5	6	Total
1	\(\frac{1}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{1}{36}\)
2	\(\frac{2}{36}\)	\(\frac{1}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{3}{36}\)
3	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{1}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{5}{36}\)
4	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{1}{36}\)	\(\frac{0}{36}\)	\(\frac{0}{36}\)	\(\frac{7}{36}\)
5	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{1}{36}\)	\(\frac{0}{36}\)	\(\frac{9}{36}\)
6	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{2}{36}\)	\(\frac{1}{36}\)	\(\frac{11}{36}\)
Totals	\(\frac{11}{36}\)	\(\frac{9}{36}\)	\(\frac{7}{36}\)	\(\frac{5}{36}\)	\(\frac{3}{36}\)	\(\frac{1}{36}\)

\(P(\{X=4\} \cap \{Y=5\}) = \frac{2}{36}\)
\(P(\{X=4\}) = \frac{5}{36}\)
\(P(\{X=2\} \cap \{Y\ge4\}) = \frac{6}{36}\)

\[\sum_{x=1}^6\sum_{y=1}^6 f_{X,Y}(x,y) = 1\]

\[\begin{split} f_X(x) & = P(X) \\ & = P(X | Y = 1) + P(X | Y = 2) + P(X | Y = 3) + P(X | Y = 4) + P(X | Y = 5) + P(X | Y = 6) \\ & = f_{X,Y}(x, 1) + f_{X,Y}(x, 2) + f_{X,Y}(x, 3) + f_{X,Y}(x, 4) + f_{X,Y}(x, 5) + f_{X,Y}(x, 6) \\ & = \sum_{y=1}^6 f_{X,Y}(x,y) \\ \end{split}\]

\[\begin{split} f_Y(y) & = P(Y) \\ & = P(Y | X = 1) + P(Y | X = 2) + P(Y | X = 3) + P(Y | X = 4) + P(Y | X = 5) + P(Y | X = 6) \\ & = f_{X,Y}(1, y) + f_{X,Y}(2, y) + f_{X,Y}(3, y) + f_{X,Y}(4, y) + f_{X,Y}(5, y) + f_{X,Y}(6, y) \\ & = \sum_{x=1}^6 f_{X,Y}(x,y) \\ \end{split}\]

EXERCISE 7

\[\begin{split} P(\{X=2\} | \{Y=3\}) = \frac{2}{5} \end{split}\]

\(x\)	1	2	3	4	5	6	Total
\(f_{X\|Y=3}(x\|y=3)\)	\(\frac{2}{5}\)	\(\frac{2}{5}\)	\(\frac{1}{5}\)	\(\frac{0}{5}\)	\(\frac{0}{5}\)	\(\frac{0}{5}\)	\(\frac{5}{5}\)

\(x\)	1	2	3	4	5	6	Total
\(f_{X\|Y=4}(x\|y=4)\)	\(\frac{2}{7}\)	\(\frac{2}{7}\)	\(\frac{2}{7}\)	\(\frac{1}{7}\)	\(\frac{0}{7}\)	\(\frac{0}{7}\)	\(\frac{7}{7}\)

\[\begin{split} f_{X|Y=y}(x|y) & = P(X | Y) \\ & = \frac{P(X \cap Y)}{P(Y)} & = \frac{f_{X,Y}(x,y)}{f_Y(y)} \\ \end{split}\]

They are not indpendent, because the value of Y affects the possible values for X. For example, if Y=4, then we know that X cannot equal 5 or 6. \[\begin{split} P(X = 3 \cap Y = 4) & = \frac{2}{7} \\ P(X = 3)*P(Y=4) & = \frac{7}{36}* \frac{7}{36} = \frac{49}{1296} \\ P(X = 3 \cap Y = 4) & \neq P(X = 3)*P(Y=4) \\ \end{split}\]