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Chapter 3: Q6E (page 117)

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

Find and sketch the p.d.f. of X

Short Answer

Expert verified

The c.d.f. of a random variable X is given as follows:

Step by step solution

01

Given the information

The c.d.f. of a random variable X is given as follows:

02

Calculating the PDF

f(x) = F鈥�(x)

f (x) = ex-3 where x鈮� 3

03

Sketch the PDF

In the graph, x-axis denotes the values of random variable x and the y-axis denotes its corresponding pdf values.

For example, for X= 2.5, PDF value = 0.60653. Plotting all the values of PDF corresponding to every values of X , the PDF curve is drawn.

Thus, the PDF curve is given by

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Most popular questions from this chapter

Question:In example 3.5.10 verify that X and Y have the same Marginal pdf and that

\({f_1}\left( x \right) = \left\{ \begin{array}{l}2k{x^2}\frac{{{{\left( {1 - {x^2}} \right)}^{\frac{2}{3}}}}}{3} for - 1 \le x \le 1\\0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\) .

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?

Let X have the uniform distribution on the interval, and let prove that \({\bf{cX + d}}\) it has a uniform distribution on the interval \(\left[ {{\bf{ca + d,cb + d}}} \right]\)

Suppose that a person鈥檚 score X on a mathematics aptitude test is a number between 0 and 1, and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in the United States, the scores X and Y are distributed according to the following joint pdf:

\(f\left( {x,y} \right)\left\{ \begin{aligned}\frac{2}{5}\left( {2x + 3y} \right)for0 \le x \le 1 and 0 \le y \le 1\\0 otherwise\end{aligned} \right.\)

a. What proportion of college students obtain a score greater than 0.8 on the mathematics test?

b. If a student鈥檚 score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?

c. If a student鈥檚 score on the mathematics test is 0.3, what is the probability that his score on the music test will be greater than 0.8?

Question:Suppose that the joint p.d.f. of X and Y is as follows:

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}24xy for x \ge 0,y \ge 0, and x + y \le 1,\\0 otherwise\end{array} \right.\).

Are X and Y independent?
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