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Chapter 3: Q2E (page 174)
Suppose that a random variable X can have each of the seven values 鈭�3, 鈭�2, 鈭�1, 0, 1, 2, 3 with equal probability. Determine the p f. Of \(Y = {X^2} - X\)
The p f of \(Y = {X^2} - X = \frac{1}{7}\)
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Question: Suppose that the joint p.d.f. of two random variablesXandYis as follows:
\(f\left( {x,y} \right) = \left\{ \begin{array}{l}c\left( {{x^2} + y} \right)\,\,\,\,for\,0 \le y \le 1 - {x^2}\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)
Determine (a) the value of the constantc;
\(\begin{array}{l}\left( {\bf{b}} \right)\,{\bf{Pr}}\left( {{\bf{0}} \le {\bf{X}} \le {\bf{1/2}}} \right){\bf{;}}\,\left( {\bf{c}} \right)\,{\bf{Pr}}\left( {{\bf{Y}} \le {\bf{X + 1}}} \right)\\\left( {\bf{d}} \right)\,{\bf{Pr}}\left( {{\bf{Y = }}{{\bf{X}}^{\bf{2}}}} \right)\end{array}\)
Suppose that the p.d.f. of a random variableXis as follows:
\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{8}x\;\;for\;0 \le x \le 4\\0\;\;\;\;otherwise\end{array} \right.\)
a. Find the value oftsuch that Pr(X鈮�t)=1/4.
b. Find the value oftsuch that Pr(X鈮�t)=1/2.
Suppose that a Markov chain has four states 1, 2, 3, 4, and stationary transition probabilities as specified by the following transition matrix
\(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{4}}&{\frac{1}{4}}&0&{\frac{1}{2}}\\0&1&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}&0\\{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}\end{array}} \right]\):
a.If the chain is in state 3 at a given timen, what is the probability that it will be in state 2 at timen+2?
b.If the chain is in state 1 at a given timen, what is the probability it will be in state 3 at timen+3?
In a large collection of coins, the probability X that a head will be obtained when a coin is tossed varies from one coin to another, and the distribution of X in the collection is specified by the following p.d.f.:
\({{\bf{f}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{6x}}\left( {{\bf{1 - x}}} \right)}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)
Suppose that a coin is selected at random from the collection and tossed once, and that a head is obtained. Determine the conditional p.d.f. of X for this coin.
If 10 percent of the balls in a certain box are red, and if 20 balls are selected from the box at random, with replacement, what is the probability that more than three red balls will be obtained?
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