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Chapter 3: Q4E (page 129)

Suppose thatXandYhave a continuous joint distribution for which the joint p.d.f. is defined as follows:

Determine (a) the value of the constant c;


Short Answer

Expert verified

a. The value of the constant is 1.5.

b. The probability is 0.375.

c. The probability is 0.125.

d. The probability is 0.50.

e. The probability is 0.00.

Step by step solution

01

Given information

The pdf of X and Y is given by,


02

Finding the value of the constant

03

Calculating the probability for part (b)

04

Calculating probability for part (c)

05

Calculating the probability for part (d)

06

Calculating the probability for part (e)

e.

Here X and Y are continuous random variables.

So the probability of continuous random variable take specific value is 0.

Pr (X = 3Y)

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Most popular questions from this chapter

Suppose that \({{\bf{X}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}\) are i.i.d. random variables andthat each of them has a uniform distribution on theinterval [0, 1]. Find the p.d.f. of\({\bf{Y = }}{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}\).

Suppose that the p.d.f. of a random variable X is as follows:

\(f\left( x \right) = \left\{ \begin{array}{l}3{x^2}\,\,for\,0 < x < 1\\0\,\,otherwise\end{array} \right.\)

Also, suppose that\(Y = 1 - {X^2}\) . Determine the p.d.f. of Y

Question:Suppose thatXandYhave a continuous joint distribution

for which the joint p.d.f. is defined as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{2}}}{{\bf{y}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\;{\bf{and}}\;{\bf{0}} \le {\bf{y}} \le {\bf{1}}\\{\bf{0}}\;\,{\bf{otherwise}}\end{array} \right.\)

a. Determine the marginal p.d.f.鈥檚 ofXandY.

b. AreXandYindependent?

c. Are the event{X<1}and the event\(\left\{ {{\bf{Y}} \ge \frac{{\bf{1}}}{{\bf{2}}}} \right\}\)independent?

Question:Suppose that two random variables X and Y have the joint p.d.f.\(f\left( {x,y} \right) = \left\{ \begin{array}{l}k{x^2}{y^2}\,\,\,\,\,\,\,\,\,\,\,\,for\,{x^2} + {y^2} \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\). Compute the conditional p.d.f. of X given

Y = y for each y.

Consider the Markov chain in Example 3.10.2 with initial

probability vector \(v = \left( {\frac{1}{2},\frac{1}{2}} \right)\) Where \(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{3}}&{\frac{2}{3}}\\{\frac{1}{3}}&{\frac{1}{3}}\end{array}} \right]\)

a.Find the probability vector specifying the probabilities

of the states at timen=2.

b.Find the two-step transition matrix
See all solutions

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