This problem does not have to have a genetics context, but we can give it one. One of the standard models for crossovers along a chromosome is that they occur as a Poisson process (rate 1). So the distances between crossovers have exponential distributions (rate parameter 1). We will have more examples on that later. For now, let's just remember the result shown in class 11/25: given a crossover occurs, its distribution is uniform on the interval.
So, think of a chromosome as the unit interval, [0,1]. Suppose we are given there were two crossovers in the segregation of this chromosome from the parent to the offspring. Let the positions be Z and W. So each of Z and W is uniformly distributed U[0,1], and they are independent.
However, when we look at the lengths of chunks of chromosome, we need to know which is smaller, Z or W. So let X=min(Z,W) be the smaller of the two, and Y=max(Z,W) be the larger. So if the parent's two chromosomes look like
--------------------------------
and ++++++++++++++++++++++++++++++++
the offspring chromosome looks like
---------++++++++++-------------
0 X Y 1
(or the mirror image
+++++++++----------+++++++++++++ )
1. Show the joint probability density (pdf) of Z and W is
fZ,W(z,w) = 1, 0 < w < 1, 0 < z < 1
Hint: think of the unit square: (Z,W) has uniform density over the
square.
2. What is P(Z < W) ?
3. Show the joint probability density (pdf) of X and Y is
fX,Y(x,y) = 2, 0 < x < y < 1
Hint: how does the unit square of Qu.1 transform to X and Y
Also, I am writing "<" since this is easier in html. All the strict
inequalities here could be <=.
4. What is the pdf of X, the length of the first chunk ?
There are at least three ways to do this.
One is to think about the pdf of Qu.3.
Since the joint pdf is the constant 2, probabilities are just 2 times area.
Hence find P(X <= x), and differentiate to get the pdf.
The second is to use the formula, integrating the joint pdf to get the
marginal pdf for X, but be careful about the limits of integration.
The third way is to go back to (Z,W), with X=min(Z,W). Note that X>x
if and only if each of Z and W is bigger than x.
5. What is the pdf of (1-Y), the length of the third chunk ?
Again there are three ways, in parallel to question 4.
6. (Harder) What is the pdf of (Y-X) the length of the middle chunk?
7. What is the conditional pdf of X, given Y=y ?
8. What is the conditional pdf of Y, given X=x ?
9. Are X and Y independent?