Chapter 3: Q1E (page 100)
Suppose that a random variableXhas the uniform distributionon the integers 10, . . . ,20. Find the probability thatXis even.
The probability that X is even is \(\frac{6}{{11}}\) .
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Suppose that\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{2}}}\)are i.i.d. random variables, and that each has the uniform distribution on the interval[0,1]. Evaluate\({\bf{P}}\left( {{{\bf{X}}_{\bf{1}}}^{\bf{2}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}^{\bf{2}} \le {\bf{1}}} \right)\)
For the conditions of Exercise 9, determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \ge {\bf{0}}{\bf{.8}}} \right)\).
Let X1…,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.
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) \({\rm P}\left( {Y > 0|x = 2} \right)\)
Question:Suppose that the joint p.d.f. ofXandYis as follows:
\(f\left( {x,y} \right) = \left\{ \begin{array}{l}2x{e^{ - y}}\;for\;0 \le x \le 1\;and\;0 < y < \infty \\0\;otherwise\end{array} \right.\)
AreXandYindependent?
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