Find the distribution of a variable from joint distribution law

probabilityprobability distributions

I'm having trouble with this particular problem.

Let the random vector $(\xi, \eta)$ has the following joint distribution law
$p(1, 1) = 1/8$, $p(2, 1) = 1/4$, $p(1, 2) = 1/8$, $p(2, 2) = 1/2$.

Find the distribution of $\xi$, if $\eta = i$, $i = 1, 2$.

What does it mean by the distribution of $\xi$? Does this mean that I have to say that

$p(\xi, 1) = 1/8 + 1/8 = 1/4$ and $p(\xi, 2) = 1/8 + 1/2 = 3/4$

I suspect that it has to do something with joint CDF, but I have no idea how to derive it from the data that I was given.
Any kind of help would be appreciated.

Best Answer

I think you are being asked to find conditional probability.

For example,

$$P(\xi=1|\eta=1)=\frac{P(\xi=1,\eta=1)}{P(\eta=1)}$$ and $$P(\xi=2|\eta=1)=\frac{P(\xi=2,\eta=1)}{P(\eta=1)}$$

Related Solutions

[Math] Joint probability of two random variables both following poisson distribution

In fact, it is always true that for two random variables $X$ and $Y$ (independent or not), $\text{Var}(\max(X,Y)) + \text{Var}(\min(X,Y)) \le \text{Var}(X) + \text{Var}(Y)$. This gives you the upper bound. I don't see an easy way to get the lower bound.

[Math] Plotting joint distribution of 2-D random variable

I typically use something like this

[X, Y] = ndgrid(-10:10, -10:10);

pdfVals = zeros(size(X));
pdfFunc = @(x,y) ... % function to evaluate the pdf
for i=1:size(X,1)
    for j=1:size(X,2)
        pdfVals(i,j) = pdfFunc(i,j);
    end
end

figure(1); clf
contour(X, Y, pdfVals);

[edit] I may have misunderstood what you have available. Given just experimental data you can show an approximate pdf with the histogram tool.

% Generate random data
nData = 1e5;
data = zeros(2,nData);
m1 = 0; m2 = 1;
s1 = 1; s2 = 2;
for i=1:nData
    d1 = m1+s1*randn;
    d2 = m2+s2*randn;
    data(:,i) = [d1; d2];
end

% hist3 will bin the data
xi = linspace(min(data(1,:)), max(data(1,:)), 50);
yi = linspace(min(data(2,:)), max(data(2,:)), 50);
hst = hist3(data',{xi yi});

% normalize the histogram data
dx = xi(2)-xi(1);
dy = yi(2)-yi(1);
area = dx*dy;
pdfData = hst/sum(sum(hst))/area;

% plot pdf
figure(2); clf
contour(xi,yi,pdfData);

Best Answer

Related Solutions

[Math] Joint probability of two random variables both following poisson distribution

[Math] Plotting joint distribution of 2-D random variable

Related Question