1. ## Markov Chain help, please!

I've been given a Markov chain problem for homework. The unfortunate part is that we will not have time to cover this material in class. So, it is learn it yourself!

Here is the initial data:

"Consider a variety of rose that can have either a pale hue or a brilliant hue. It is known that seeds from a pale blossom yield plants of which 60% have pale flowers and 40% have brilliant flowers. Seeds from a brilliant blossom yield plants of which 30% have pale and 70% have brilliant flowers."

This is what I am stuck on:

"Now, let p and b represent the proportion of plants with pale and brilliant hues, respectively, and suppose the proportions do stabilize. Then, the proportion of flowers in the next generation that are pale is .6p + .3b. (Why?) But since the population has stabilized, the proportion of pale flowers in the next generation must remain the same as it was. Write an equation to represent this situation:

Similarly, the proportion of flowers that are brilliant in the next generation after the population has stabilized will remain the same. Write an equation to represent the situation for the brilliant flowers."

If anyone can help me understand this, I would greatly appreciate it! It is due on Tuesday evening, and I've been working on this for over a week trying to figure it out myself. Thank you!

2. Originally Posted by supercanuck
I've been given a Markov chain problem for homework. The unfortunate part is that we will not have time to cover this material in class. So, it is learn it yourself!

Here is the initial data:

"Consider a variety of rose that can have either a pale hue or a brilliant hue. It is known that seeds from a pale blossom yield plants of which 60% have pale flowers and 40% have brilliant flowers. Seeds from a brilliant blossom yield plants of which 30% have pale and 70% have brilliant flowers."

This is what I am stuck on:

"Now, let p and b represent the proportion of plants with pale and brilliant hues, respectively, and suppose the proportions do stabilize. Then, the proportion of flowers in the next generation that are pale is .6p + .3b. (Why?) But since the population has stabilized, the proportion of pale flowers in the next generation must remain the same as it was. Write an equation to represent this situation:

Similarly, the proportion of flowers that are brilliant in the next generation after the population has stabilized will remain the same. Write an equation to represent the situation for the brilliant flowers."

If anyone can help me understand this, I would greatly appreciate it! It is due on Tuesday evening, and I've been working on this for over a week trying to figure it out myself. Thank you!
We have to suppose that the number of seeds does not depend on the kind of hue. Then, at the next generation, there are the pale flowers produced by pale flowers, the pale flowers produced by brilliant flowers, the brilliant flowers produced by pale flowers and the brilliant flowers produced by brilliant flowers.
The proportion of the first kind of flower is given by the product of the proportion of pale flowers in the first generation by the proportion of pale flowers that one pale flower produces. That gives $\displaystyle 0.6p$. And it is the same for the other ones.
From there, you can deduce the equation that is given. And the stationarity just gives $\displaystyle p=0.6p+0.3b$ (the proportion remains the same), and $\displaystyle b=0.4p+0.7b$ for the brilliant flowers. Since $\displaystyle p+b=1$, you can then compute $\displaystyle p$ and $\displaystyle b$ (using any of the two previous equations).

3. ## Thank you!

That was what I was heading towards, but because I haven't learned anything about Markov chains, I was not 100% sure. You have been very helpful!

4. Hello, supercanuck!

I think I can summarize all this data for you . . .

Consider a variety of rose that can have either a pale hue or a bright hue.
Seeds from a pale rose yield plants of which 60% have pale flowers and 40% have bright flowers.
Seeds from a bright rose yield plants of which 30% have pale and 70% bright flowers.

Now, let $\displaystyle p$ and $\displaystyle b$ represent the proportion of plants with pale and bright hues, respectively,
and suppose the proportions do stabilize.

Then, the proportion of flowers in the next generation that are pale is 0.6p + 0.3b. (Why?)
But since the population has stabilized, the proportion of pale flowers in the next generation
must remain the same as it was. Write an equation to represent this situation.

Similarly, the proportion of flowers that are bright in the next generation
after the population has stabilized will remain the same.
Write an equation to represent the situation for the bright flowers.

We are given this data:

. . $\displaystyle \begin{array}{c|c|c} & \text{pale} & \text{bright} \\ \hline \hline \text{Pale} & 0.6 & 0.4 \\ \hline \text{Bright} & 0.3 & 0.7 \\ \hline\end{array}$

The transition matrix is: .$\displaystyle A \:=\:\begin{bmatrix}0.6 & 0.4 \\ 0.3 & 0.7 \end{bmatrix}$

We have: .$\displaystyle G \:=\:[p\;\;b]$ as the percent of pale and bright roses, respectively,
. . for a particular generation of roses.

Then $\displaystyle GA$ gives the percent of pale/bright for the next generation.

That is: .$\displaystyle GA \:=\:\begin{bmatrix}p & b\end{bmatrix}\begin{bmatrix} 0.6 & 0.4\\0.3&0.7 \end{bmatrix} \;=\;\begin{bmatrix}\underbrace{0.6p + 0.3b} & \underbrace{0.4p + 0.7b} \end{bmatrix}$
. . . . . . . . . . . . . . . . . . . . . . . . . . $\displaystyle ^{\text{pale}} \qquad\qquad ^{\text{bright}}$

5. ## Thank you!

This clears up even more of the Markov mystery! It is rather difficult trying to learn this stuff without proper instruction, so thank you for your help!