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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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Suppose that a random variableXhas the uniform distribution on the interval [鈭�2,8]. Find the p.d.f. ofXand the value of Pr(0<X <7).
Suppose that the joint p.d.f. of two random variables X and Y is as follows:
\(f\left( {x,y} \right) = \left\{ \begin{aligned}{l}c\left( {x + {y^2}} \right)\,\,\,\,\,\,for\,0 \le x \le 1\,and\,0 \le y \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{aligned} \right.\)
Determine
(a) the conditional p.d.f. of X for every given value of Y, and
(b) \({\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{2}} \right)\).
Let X1鈥�,Xn be independent random variables, and let W be a random variable such that \({\rm P}\left( {w = c} \right) = 1\) for some constant c. Prove that \({x_1},....,{x_n}\)they are conditionally independent given W = c.
An insurance agent sells a policy that has a \(100 deductible
and a \)5000 cap. When the policyholder files a claim, the policyholder must pay the first \(100. After the first \)100, the insurance company pays therest of the claim up to a maximum payment of $5000. Any
excess must be paid by the policyholder. Suppose that thedollar amount X of a claim has a continuous distribution
with p.d.f. \({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\frac{{\bf{1}}}{{{{\left( {{\bf{1 + x}}} \right)}^{\bf{2}}}}}\) for x>0 and 0 otherwise.
LetY be the insurance company's amount to payon the claim.
a. Write Y as a function of X, i.e., \({\bf{Y = r}}\left( {\bf{X}} \right).\)
b. Find the c.d.f. of Y.
c. Explain why Y has neither a continuous nor a discretedistribution.
Suppose that\({X_1}....{X_n}\)are i.i.d. random variables, each having the following c.d.f.:\(F\left( x \right) = \left\{ \begin{array}{l}0\,\,\,\,\,\,\,\,\,\,\,\,\,\,for\,x \le 0\\1 - {e^{ - x}}\,\,\,for\,x > 0\end{array} \right.\)
Let\({Y_1} = min\left\{ {{X_1},{X_2}..{X_n}} \right\}\)and\({Y_n} = max\left\{ {{X_{1,}}{X_2}..{X_n}} \right\}\)Determine the conditional p.d.f. of\({Y_1}\)given that\({Y_n} = {y_n}\)
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