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Chapter 3: Q15E (page 93)

Verify the rows of the transition matrix in Example 3.10.6 that correspond to current states\(\left\{ {AA,Aa} \right\}\)and\(\left\{ {Aa,aa} \right\}\)

Short Answer

Expert verified

Rows of transition matrix that correspond to current states \(\left\{ {AA,Aa} \right\}\) and \(\left\{ {Aa,aa} \right\}\)

Step by step solution

01

Computing the transition matrix of the process of the second cross.

Consider a make and a female plant both with allele Aa. First, cross the two plants with genotypes Aa and aa to produce 2 offspring.

After the cross, the parent plants are destroyed. The offspring of the cross are further crossed with each other. There are four possible ways to combine genes in the offspring.

An offspring could get a from the ovule and a from the pollen, a from the ovule and A from pollen, A from the ovule and a from the pollen, A from the ovule, and A from the pollen.

The possibility of each way has a possibility of 0.25.

Hence the transition matrix can be listed in the table as follows:

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Most popular questions from this chapter

Let Y be the rate (calls per hour) at which calls arrive at a switchboard. Let X be the number of calls during a two-hour period. Suppose that the marginal p.d.f. of Y is

\({{\bf{f}}_{\bf{2}}}\left( {\bf{y}} \right){\bf{ = }}\left\{ {\begin{align}{}{{{\bf{e}}^{{\bf{ - y}}}}}&{{\bf{if}}\,{\bf{y > 0,}}}\\{\bf{0}}&{{\bf{otherwise,}}}\end{align}} \right.\)

And that the conditional p.d.f. of X given\({\bf{Y = y}}\)is

\({{\bf{g}}_{\bf{1}}}\left( {{\bf{x}}\left| {\bf{y}} \right.} \right){\bf{ = }}\left\{ {\begin{align}{}{\frac{{{{\left( {{\bf{2y}}} \right)}^{\bf{x}}}}}{{{\bf{x!}}}}{{\bf{e}}^{{\bf{ - 2y}}}}}&{{\bf{if}}\,{\bf{x = 0,1,}}...{\bf{,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\)

  1. Find the marginal p.d.f. of X. (You may use the formula\(\int_{\bf{0}}^\infty {{{\bf{y}}^{\bf{k}}}{{\bf{e}}^{{\bf{ - y}}}}{\bf{dy = k!}}} {\bf{.}}\))
  2. Find the conditional p.d.f.\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{0}} \right.} \right)\)of Y given\({\bf{X = 0}}\).
  3. Find the conditional p.d.f.\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| 1 \right.} \right)\)of Y given\({\bf{X = 1}}\).
  4. For what values of y is\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{1}} \right.} \right){\bf{ > }}{{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{0}} \right.} \right)\)? Does this agree with the intuition that the more calls you see, the higher you should think the rate is?

Suppose that an electronic system comprises four components, and let\({X_j}\)denote the time until component j fails to operate (j = 1, 2, 3, 4). Suppose that\({X_1},{X_2},{X_3}\)and\({X_4}\)are i.i.d. random variables, each of which has a continuous distribution with c.d.f.\(F\left( x \right)\)Suppose that the system will operate as long as both component 1 and at least one of the other three components operate. Determine the c.d.f. of the time until the system fails to operate.

Suppose that a fair coin is tossed 10 times independently.

Determine the p.f. of the number of heads that will be obtained.

Suppose that the joint distribution of X and Y is uniform over a set A in the xy-plane. For which of the following sets A are X and Y independent?

a. A circle with a radius of 1 and with its center at the origin

b. A circle with a radius of 1 and with its center at the point (3,5)

c. A square with vertices at the four points (1,1), (1,鈭�1), (鈭�1,鈭�1), and (鈭�1,1)

d. A rectangle with vertices at the four points (0,0), (0,3), (1,3), and (1,0)

e. A square with vertices at the four points (0,0), (1,1),(0,2), and (鈭�1,1)

Suppose that a point (X, Y) is chosen at random from the disk S defined as follows:

\(S = \left\{ {\left( {x,y} \right) :{{\left( {x - 1} \right)}^2} + {{\left( {y + 2} \right)}^2} \le 9} \right\}.\) Determine (a) the conditional pdf of Y for every given value of X, and (b) \({\rm P}\left( {Y > 0|x = 2} \right)\)
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