16  Bivariate Probability

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16.1 Theoretical Distributions

We now consider a bivariate random vector \((X_{i}, Y_{i})\), which is a theoretical version of the bivariate observations \((\hat{X}_{i}, \hat{Y}_{i})\). E.g., If we are going to flip two coins, then \((X_{i}, Y_{i})\) corresponds to the unflipped coins and \((\hat{X}_{i}, \hat{Y}_{i})\) corresponds to concrete values after they are flipped.

Definitions for Discrete Data.

The joint distribution is defined as \[\begin{eqnarray} Prob(X_{i} = x, Y_{i} = y) \end{eqnarray}\]

Note that variables are statistically independent if \(Prob(X_{i} = x, Y_{i} = y)= Prob(X_{i} = x) Prob(Y_{i} = y)\) for all \(x, y\). Independence is sometimes assumed for mathematical simplicity, not because it generally fits data well.¹

The marginal distributions are then defined as \[\begin{eqnarray} Prob(X_{i} = x) = \sum_{y} Prob(X_{i} = x | Y_{i} = y) Prob( Y_{i} = y ) \\ Prob(Y_{i} = y) = \sum_{x} Prob(Y_{i} = y | X_{i} = x) Prob( X_{i} = x ), \end{eqnarray}\] which is also known as the law of total probability.

The conditional distributions are defined as \[\begin{eqnarray} Prob(X_{i} = x | Y_{i} = y) = \frac{ Prob(X_{i} = x, Y_{i} = y)}{ Prob( Y_{i} = y )}\\ Prob(Y_{i} = y | X_{i} = x) = \frac{ Prob(X_{i} = x, Y_{i} = y)}{ Prob( X_{i} = x )} \end{eqnarray}\]

Coin Flips Example.

For one example, Consider flipping two coins, where we mark whether “heads” is face up with a \(1\) and “tail” with a \(0\). E.g., the first coin has a value of \(x=1\) if it shows heads and \(x=0\) if it shows tails. This table shows both the joint distribution and also each marginal distribution.

	\(x=0\)	\(x=1\)	Marginal
\(y=0\)	\(Prob(X_{i}=0,Y_{i}=0)\)	\(Prob(X_{i}=1,Y_{i}=0)\)	\(Prob(Y_{i}=0)\)
\(y=1\)	\(Prob(X_{i}=0,Y_{i}=1)\)	\(Prob(X_{i}=1,Y_{i}=1)\)	\(Prob(Y_{i}=1)\)
Marginal	\(Prob(X_{i}=0)\)	\(Prob(X_{i}=1)\)	\(1\)

Note that different joint distributions can have the same marginal distributions.

Note

Suppose both coins are “fair”: \(Prob(X_{i}=1)= 1/2\) and \(Prob(Y_{i}=1|X_{i}=x)=1/2\) for either \(x=1\) or \(x=0\), then the four potential outcomes have equal probabilities. \[\begin{eqnarray} Prob(X_{i} = 0, Y_{i} = 0) &=& 1/2 \times 1/2 = 1/4 \\ Prob(X_{i} = 0, Y_{i} = 1) &=& 1/4 \\ Prob(X_{i} = 1, Y_{i} = 0) &=& 1/4 \\ Prob(X_{i} = 1, Y_{i} = 1) &=& 1/4 . \end{eqnarray}\] The joint distribution is written generally as \[\begin{eqnarray} Prob(X_{i} = x, Y_{i} = y) &=& Prob(X_{i} = x) Prob(Y_{i} = y). \end{eqnarray}\]

# Create a 2x2 matrix for the joint distribution.
# Rows correspond to X (coin 1), and columns correspond to Y (coin 2).
probs <- c(0.25, 0.25, 0.25, 0.25) # Fair coin
P_xy <- matrix( probs, nrow = 2, ncol = 2)
rownames(P_xy) <- c("Y=0", "Y=1")
colnames(P_xy) <- c("X=0", "X=1")
P_xy
##      X=0  X=1
## Y=0 0.25 0.25
## Y=1 0.25 0.25

# Draw N bivariate observations
N <- 500
xy_vals <- expand.grid(y = 0:1, x = 0:1)
draws <- sample(nrow(xy_vals), N, replace = TRUE, prob = P_xy)
x_draw <- xy_vals[draws, "x"]
y_draw <- xy_vals[draws, "y"]

# With enough draws, empirical frequencies approach the true probabilities
table(y_draw, x_draw) / N
##       x_draw
## y_draw     0     1
##      0 0.246 0.276
##      1 0.248 0.230
P_xy
##      X=0  X=1
## Y=0 0.25 0.25
## Y=1 0.25 0.25

The marginal distribution of the second coin is \[\begin{eqnarray} Prob(Y_{i} = 0) &=& Prob(Y_{i} = 0 | X_{i} = 0) Prob(X_{i}=0) + Prob(Y_{i} = 0 | X_{i} = 1) Prob(X_{i}=1)\\ &=& 1/2 (1/2) + 1/2 (1/2) = 1/2\\ Prob(Y_{i} = 1) &=& Prob(Y_{i} = 1 | X_{i} = 0) Prob(X_{i}=0) + Prob(Y_{i} = 1 | X_{i} = 1) Prob(X_{i}=1)\\ &=& 1/2 (1/2) + 1/2 (1/2) = 1/2 \end{eqnarray}\]

The marginal distribution of the first coin is found in the exact same way \[\begin{eqnarray} Prob(X_{i} = 0) &=& Prob(X_{i} = 0 | Y_{i} = 0) Prob(Y_{i}=0) + Prob(X_{i} = 0 | Y_{i} = 1) Prob(Y_{i}=1)\\ &=& 1/2 (1/2) + 1/2 (1/2) = 1/2\\ Prob(X_{i} = 1) &=& Prob(X_{i} = 1 | Y_{i} = 0) Prob(Y_{i}=0) + Prob(X_{i} = 1 | Y_{i} = 1) Prob(Y_{i}=1)\\ &=& 1/2 (1/2) + 1/2 (1/2) = 1/2 \end{eqnarray}\]

Altogether, we find

	\(x=0\)	\(x=1\)	Marginal
\(y=0\)	\(1/4\)	\(1/4\)	\(1/2\)
\(y=1\)	\(1/4\)	\(1/4\)	\(1/2\)
Marginal	\(1/2\)	\(1/2\)	\(1\)

# Compute the marginal distributions.
# Marginal for X: sum across columns.
P_x <- colSums(P_xy)
P_x
## X=0 X=1 
## 0.5 0.5
# Marginal for Y: sum across rows.
P_y <- rowSums(P_xy)
P_y
## Y=0 Y=1 
## 0.5 0.5

# Compute the conditional probabilities Prob(Y | X).
PY_x <- matrix(0, nrow = 2, ncol = 2)
for (j in c(1,2)) {
  PY_x[, j] <- P_xy[, j] / P_x[j]
}
rownames(PY_x) <- c("Y=0", "Y=1")
colnames(PY_x) <- c("given X=0", "given X=1")
PY_x
##     given X=0 given X=1
## Y=0       0.5       0.5
## Y=1       0.5       0.5

Are the coins statistically independent?

Note

Now consider a second example, where the second coin is “Completely Unfair”, so that it is always the same as the first. The outcomes generated with a Completely Unfair coin are the same as if we only flipped one coin. \[\begin{eqnarray} Prob(X_{i} = 0, Y_{i} = 0) &=& 1/2 \\ Prob(X_{i} = 0, Y_{i} = 1) &=& 0 \\ Prob(X_{i} = 1, Y_{i} = 0) &=& 0 \\ Prob(X_{i} = 1, Y_{i} = 1) &=& 1/2 . \end{eqnarray}\] The joint distribution is written generally as \[\begin{eqnarray} Prob(X_{i} = x, Y_{i} = y) &=& Prob(X_{i} = x) \mathbf{1}( x=y ), \end{eqnarray}\] where \(\mathbf{1}(X_{i}=1)\) means \(X_{i}= 1\) and \(0\) if \(X_{i}\neq0\). The marginal distribution of the second coin is \[\begin{eqnarray} Prob(Y_{i} = 0) &=& Prob(Y_{i} = 0 | X_{i} = 0) Prob(X_{i}=0) + Prob(Y_{i} = 0 | X_{i} = 1) Prob(X_{i} = 1)\\ &=& 1 (1/2) + 0(1/2) = 1/2 .\\ Prob(Y_{i} = 1) &=& Prob(Y_{i} = 1 | X_{i} =0) Prob( X_{i} = 0) + Prob(Y_{i} = 1 | X_{i} = 1) Prob( X_{i} = 1)\\ &=& 0 (1/2) + 1 (1/2) = 1/2 . \end{eqnarray}\] which is the same marginal as in the first example!

The marginal distribution of the first coin is found in the exact same way (show this yourself).

Alltogether, we find

	\(x=0\)	\(x=1\)	Marginal
\(y=0\)	\(1/2\)	\(0\)	\(1/2\)
\(y=1\)	\(0\)	\(1/2\)	\(1/2\)
Marginal	\(1/2\)	\(1/2\)	\(1\)

# Create the joint distribution matrix for the unfair coin case.
probs <- c(0.5, 0, 0, 0.5) # unfair coin

P_xy <- matrix( probs, nrow = 2, ncol = 2)
rownames(P_xy) <- c("Y=0", "Y=1")
colnames(P_xy) <- c("X=0", "X=1")
P_xy
##     X=0 X=1
## Y=0 0.5 0.0
## Y=1 0.0 0.5


# Compute the marginal distributions.
# Marginal for X: sum across columns.
P_x <- colSums(P_xy)
P_x
## X=0 X=1 
## 0.5 0.5
# Marginal for Y: sum across rows.
P_y <- rowSums(P_xy)
P_y
## Y=0 Y=1 
## 0.5 0.5

# Compute the conditional probabilities Prob(Y | X).
PY_x <- matrix(0, nrow = 2, ncol = 2)
for (j in c(1,2)) {
  PY_x[, j] <- P_xy[,j] / P_x[j]
}
rownames(PY_x) <- c("Y=0", "Y=1")
colnames(PY_x) <- c("given X=0", "given X=1")
PY_x
##     given X=0 given X=1
## Y=0         1         0
## Y=1         0         1

Are the coins statistically independent?

Tip

Try computing conditional probabilities for another unfair set of coins below. Explain why the marginal distribution of the second coin is balanced.

probs <- c(0.4, 0.2, 0.1, 0.3)
P_xy <- matrix(probs, nrow = 2,
  dimnames = list(c("Y=0","Y=1"), c("X=0","X=1")))

# Draw N bivariate observations
xy_vals <- expand.grid(y = 0:1, x = 0:1)
N <- 500
draws <- sample(nrow(xy_vals), N, replace = TRUE, prob = P_xy)
x_draw <- xy_vals[draws, "x"]
y_draw <- xy_vals[draws, "y"]

table(y_draw, x_draw) / N
##       x_draw
## y_draw     0     1
##      0 0.414 0.100
##      1 0.208 0.278
P_xy
##     X=0 X=1
## Y=0 0.4 0.1
## Y=1 0.2 0.3

Definitions for Continuous Data.

The joint distribution is defined as \[\begin{eqnarray} F(x, y) &=& Prob(X_{i} \leq x, Y_{i} \leq y) \end{eqnarray}\] The marginal distributions are then defined as \[\begin{eqnarray} F_{X}(x) &=& F(x, \infty)\\ F_{Y}(y) &=& F(\infty, y). \end{eqnarray}\] which is also known as the law of total probability. Variables are statistically independent if \(F(x, y) = F_{X}(x)F_{Y}(y)\) for all \(x, y\).

For example, suppose \((X_{i},Y_{i})\) is bivariate normal with means \((\mu_{X}, \mu_{Y})\), variances \((\sigma_{X}^2, \sigma_{Y}^2)\) and correlation \(\rho\).

library(mvtnorm)

# Simulate Bivariate Data
N <- 10000
Mu <- c(2,2) ## Means

Sigma1 <- matrix(c(2,-.8,-.8,1),2,2) ## CoVariance Matrix 1
MVdat1 <- rmvnorm(N, Mu, Sigma1)
colnames(MVdat1) <- c('X','Y')

Sigma2 <- matrix(c(2,.4,.4,1),2,2) ## CoVariance Matrix 2
MVdat2 <- rmvnorm(N, Mu, Sigma2)
colnames(MVdat2) <- c('X','Y')

par(mfrow=c(1,2))
## Different diagonals
plot(MVdat2, col=rgb(1,0,0,0.02), pch=16,
    main='Joint Distributions', font.main=1,
    ylim=c(-4,8), xlim=c(-4,8),
    xlab='X', ylab='Y')
points(MVdat1,col=rgb(0,0,1,0.02),pch=16)
## Same marginal distributions
xbks <- seq(-4,8,by=.2)
hist(MVdat2[,2], col=rgb(1,0,0,0.5),
    breaks=xbks, border=NA,
    xlab='Y',
    main='Marginal Distributions', font.main=1)
hist(MVdat1[,2], col=rgb(0,0,1,0.5),
    add=T, breaks=xbks, border=NA)

# See that independent data are a special case
n <- 2e4
## 2 Indepenant RV
XYiid <- cbind( rnorm(n),  rnorm(n))
## As a single Joint Draw
XYjoint <- mvtnorm::rmvnorm(n, c(0,0))
## Plot
par(mfrow=c(1,2))
plot(XYiid, xlab=
    col=grey(0,.05), pch=16, xlim=c(-5,5), ylim=c(-5,5))
plot(XYjoint,
    col=grey(0,.05), pch=16, xlim=c(-5,5), ylim=c(-5,5))

# Compare densities
#d1 <- dnorm(XYiid[,1],0)*dnorm(XYiid[,2],0)
#d2 <- mvtnorm::dmvnorm(XYiid, c(0,0))
#head(cbind(d1,d2))

The multivariate normal is a workhorse for analytical work on multivariate random variables, but there are many more. See e.g., https://cran.r-project.org/web/packages/NonNorMvtDist/NonNorMvtDist.pdf

16.2 Statistics

Conditional Expectation.

The conditional expectation function \(\mathbb{E}[Y_i|X_i=x]\) is the average value of \(Y_i\) among observations with \(X_i=x\). In empirical work, this is the population object that regressions try to approximate.

For discrete \(X_i\), the conditional expectation function is a weighted average over conditional probabilities: \[\begin{eqnarray} \mathbb{E}[Y_i|X_i=x] = \sum_y y Prob(Y_i=y|X_i=x). \end{eqnarray}\] For continuous \(X_i\), the conditional expectation function is a smooth curve indexed by \(x\).

For example, consider a bivariate random vector with three outcomes for \(X_{i}\) and three outcomes for \(Y_{i}\).

	\(x=0\)	\(x=1\)	\(x=2\)
\(y=0\)	\(0.0\)	\(0.1\)	\(0.0\)
\(y=10\)	\(0.1\)	\(0.3\)	\(0.1\)
\(y=20\)	\(0.1\)	\(0.1\)	\(0.2\)

The plot below shows each outcome \((x,y)\) with deeper colors reflecting higher probability events.

# Make a Probability Table
x <- c(0,1,2)
y <- c(0,10,20)
P_xy <- matrix(c(
    0.0, 0.1, 0.0,
    0.1, 0.3, 0.1,
    0.1, 0.1, 0.2
), nrow=3, ncol=3, byrow=TRUE)
rownames(P_xy) <- paste0('y=',y)
colnames(P_xy) <-  paste0('x=',x)
P_xy
##      x=0 x=1 x=2
## y=0  0.0 0.1 0.0
## y=10 0.1 0.3 0.1
## y=20 0.1 0.1 0.2

# Plot joint distribution
prob_table <- expand.grid(x=x, y=y)
prob_table[,'probabilities'] <- as.vector(t(P_xy))
plot(prob_table[,'x'], prob_table[,'y'],
     xlim=c(-0.5,2.5), ylim=c(-5, 25),
     pch=21, cex=8, col='blue',
     bg=rgb(0,0,1, prob_table[,'probabilities']),
     xlab = "x", ylab = "y")

We can compute marginals from the table, \[\begin{eqnarray} Prob(X_{i}=0)=0.2,\quad & Prob(X_{i}=1)=0.5 &,\quad Prob(X_{i}=2) = 0.3 \\ Prob(Y_{i}=0)=0.1,\quad & Prob(Y_{i}=10)=0.5 &,\quad Prob(Y_{i}=20) = 0.4 \end{eqnarray}\]

This gives us conditional probabilities and hence also conditional expectations; \[\begin{eqnarray} \mathbb{E}[Y_i|X_i=0] &=& 0\frac{0}{0.2} + 10\frac{0.1}{0.2} + 20\frac{0.1}{0.2} = 0 + \frac{10}{2} + \frac{20}{2} = 15,\\ \mathbb{E}[Y_i|X_i=1] &=& 0\frac{0.1}{0.5} + 10\frac{0.3}{0.5} + 20\frac{0.1}{0.5} = 0 + \frac{30}{5} + \frac{20}{5} = 10,\\ \mathbb{E}[Y_i|X_i=2] &=& 0\frac{0}{0.3} + 10\frac{0.1}{0.3} + 20\frac{0.2}{0.3} = 0 + \frac{10}{3} + \frac{40}{3} = \frac{50}{3} \approx 16.67. \end{eqnarray}\] Note that the conditional mean is non-monotonic: \[\begin{eqnarray} \frac{\mathbb{E}[Y_i|X_i=1] - \mathbb{E}[Y_i|X_i=0]}{1-0} &=& \frac{10 - 15}{1} = -5,\\ \frac{\mathbb{E}[Y_i|X_i=2] - \mathbb{E}[Y_i|X_i=1]}{2-1} &=& \frac{\frac{50}{3} - 10}{1} = \frac{20}{3} \approx 6.67. \end{eqnarray}\] It decreases from \(x=0\) to \(x=1\) and then increases from \(x=1\) to \(x=2\).

# Compute Marginals
P_x <- colSums(P_xy) #Prob(X_i=x)
P_y <- rowSums(P_xy) #Prob(Y_i=y)

# Compute the conditional probabilities Prob(Y_i=y | X_i=x).
PY_x <- matrix(0, nrow = 3, ncol = 3)
for (j in seq(ncol(P_xy)) ) {
  PY_x[, j] <- P_xy[,j] / P_x[j]
}
rownames(PY_x) <- paste0("Y=", y)
colnames(PY_x) <- paste0("given X=", x)
PY_x
##      given X=0 given X=1 given X=2
## Y=0        0.0       0.2 0.0000000
## Y=10       0.5       0.6 0.3333333
## Y=20       0.5       0.2 0.6666667

# Computational Shortcut: PY_x <- t( t(P_xy) / P_x )

# Conditional expectation E[Y_i|X_i=x]
EY_x <- colSums(PY_x * y)
EY_x
## given X=0 given X=1 given X=2 
##  15.00000  10.00000  16.66667

# Contrast with Grand Means
EX <- sum(x * P_x)
EY <- sum(y * P_y)
EY
## [1] 13

# E[X_i|Y_i=y]
# PX_y <- P_xy / P_y  #Prob(X_i=x | Y_i=y)
# EX_y <- colSums(PX_y * x)

Covariance.

We can also dig a little deeper into the other bivariate statistics we compute. When we know how the data are generated, we can often compute the foundational covariance statistic. Referring to and \(\mu_{X}=\mathbb{E}[X_{i}]\) and \(\mu_{Y}=\mathbb{E}[Y_{i}]\), we have \[\begin{eqnarray} \mathbb{C}[X_{i}, Y_{i}] &=& \mathbb{E}[(X_{i} – \mu_{X})(Y_{i} – \mu_{Y}])] = \sum_{x}\sum_{y} (x – \mu_{X})(y – \mu_{Y}) Prob(X_{i} = x, Y_{i} = y) \end{eqnarray}\]

Using the marginal distributions from before allows us to compute the means: \[\begin{eqnarray} \mathbb{E}[X_{i}] &=& 0(0.2)+1(0.5)+2(0.3) = 1.1 \\ \mathbb{E}[Y_{i}] &=& 0(0.1)+10(0.5)+20(0.4) = 13 \end{eqnarray}\] We can then compute the cell-by-cell contributions: \(Prob(X_{i} = x, Y_{i} = y) (x-\mathbb{E}[X_{i}])(y-\mathbb{E}[Y_{i}])\), which lead plug in to the covariance formula; \[\begin{eqnarray} \begin{array}{l l r r r r r} \hline x & y & Prob(X_{i}=x, Y_{i}=y) & x-\mathbb{E}[X_{i}] & y-\mathbb{E}[Y_{i}] & (x-\mathbb{E}[X_{i}])(y-\mathbb{E}[Y_{i}]) & \text{Contribution}\\ \hline 0 & 0 & 0.0 & -1.1 & -13 & 14.3 & 0\\ 0 & 10 & 0.1 & -1.1 & -3 & 3.3 & 0.330\\ 0 & 20 & 0.1 & -1.1 & 7 & -7.7 & -0.770\\ 1 & 0 & 0.1 & -0.1 & -13 & 1.3 & 0.130\\ 1 & 10 & 0.3 & -0.1 & -3 & 0.3 & 0.090\\ 1 & 20 & 0.1 & -0.1 & 7 & -0.7 & -0.070\\ 2 & 0 & 0.0 & 0.9 & -13 & -11.7 & 0\\ 2 & 10 & 0.1 & 0.9 & -3 & -2.7 & -0.270\\ 2 & 20 & 0.2 & 0.9 & 7 & 6.3 & 1.260\\ \hline \end{array} \end{eqnarray}\] \[\begin{eqnarray} \mathbb{C}[X_{i},Y_{i}] &=& \sum_{x} \sum_{y} \left(x-\mathbb{E}[X_{i}]\right)\left(y-\mathbb{E}[Y_{i}]\right) Prob\left(X_{i} = x, Y_{i} = y\right) \\ &=& 0 + 0.330 -0.770 + 0.130 + 0.090 -0.070 +0 -0.270 + 1.260 = 0.7 \end{eqnarray}\]

Correlation

We can not compute the often-used the Pearson correlation: \[\begin{eqnarray} \frac{\mathbb{C}[X_{i}, Y_{i}] }{ \sqrt{\mathbb{V}[X_{i}]} \sqrt{\mathbb{V}[Y_{i}]} } \end{eqnarray}\]

To compute this statistic in the previous example, we first need the standard deviations \[\begin{eqnarray} \mathbb{V}[X_{i}] &=& \sum_{x} (x-\mathbb{E}[X_{i}])^2 Prob(X_{i} = x) \\ &=& (0-1.1)^2(0.2)+(1-1.1)^2(0.5)+(2-1.1)^2(0.3)=0.49 \\ \mathbb{V}[Y_{i}] &=& \sum_{y} (y-\mathbb{E}[Y_{i}])^2 Prob(Y_{i} = y) \\ &=& (0-13)^2(0.1)+(10-13)^2(0.5)+(20-13)^2(0.4)=41 \\ \end{eqnarray}\] Then we can find the correlation as \[\begin{eqnarray} \frac{\mathbb{C}[X_{i},Y_{i}]}{ \sqrt{\mathbb{V}[X_{i}]} \sqrt{\mathbb{V}[Y_{i}]} } &=& \frac{0.7}{\sqrt{0.49} \sqrt{41}} \approx 0.156, \end{eqnarray}\] which suggests a weak positive association between the variables.

Note that you can do all of the above calculations using the computer instead of by hand.

# Compute Covariance
dxy_grid <- expand.grid(dy=y-EY, dx=x-EX)[,c(2,1)]
dxy_grid[,'p'] <- as.vector(P_xy)
dxy_grid[,'contribution'] <- dxy_grid[,'dx'] * dxy_grid[,'dy'] * dxy_grid[,'p']
CovXY <- sum(dxy_grid[,'contribution'])
CovXY
## [1] 0.7

# Compute Variances
VX <- sum( (x-EX)^2 * P_x)
SX <- sqrt(VX)
VY <- sum( (y-EY)^2 * P_y)
SY <- sqrt(VY)

# Compute Correlation
CorXY <- CovXY / (SX * SY)
CorXY
## [1] 0.1561738

Note

Compute the correlation for bivariate data with these probabilities, using both math (first) and the computer (second)

	\(x=-10\)	\(x=10\)
\(y=0\)	\(0.05\)	\(0.20\)
\(y=1\)	\(0.05\)	\(0.20\)
\(y=2\)	\(0.05\)	\(0.20\)
\(y=3\)	\(0.05\)	\(0.20\)

Tip

Also compute the correlation for bivariate data with these probabilities

	\(x=-10\)	\(x=10\)
\(y=0\)	\(0.05\)	\(0.05\)
\(y=1\)	\(0.10\)	\(0.10\)
\(y=2\)	\(0.15\)	\(0.15\)
\(y=3\)	\(0.20\)	\(0.20\)

Also compute the correlation for bivariate data with these probabilities

	\(x=-10\)	\(x=10\)
\(y=0\)	\(0.05\)	\(0.15\)
\(y=1\)	\(0.05\)	\(0.15\)
\(y=2\)	\(0.10\)	\(0.20\)
\(y=3\)	\(0.10\)	\(0.20\)

Explain intuitively when the correlation equals \(0\) and when it does not.

Regression Coefficient

Similarly, we can also compute the regression coefficient \[\begin{eqnarray} \frac{\mathbb{C}[X_{i}, Y_{i}] }{ \mathbb{V}[X_{i}] } \end{eqnarray}\]

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1. The same can be said about assuming normally distributed errors, although at least that can be motivated by the Central Limit Theorems.