Chapter 3: Q4E (page 116)
Suppose that the c.d.f.Fof a random variableXis as sketched in Fig. 3.9. Find each of the following probabilities:


Short Answer
a.0.1.
b.0.1.
c.0.2
d.0
e.0.6
f.0.4
g.0.7
h.0
i.0
j.0
k.0
l.0.2
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Chapter 3: Q4E (page 116)
Suppose that the c.d.f.Fof a random variableXis as sketched in Fig. 3.9. Find each of the following probabilities:


a.0.1.
b.0.1.
c.0.2
d.0
e.0.6
f.0.4
g.0.7
h.0
i.0
j.0
k.0
l.0.2
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Determine the p.d.f. of the range of a random sample of n observations from the uniform distribution on the interval [−3, 5].
Question:Prove Theorem 3.5.6.
Let X and Y have a continuous joint distribution. Suppose that
\(\;\left\{ {\left( {x,y} \right):f\left( {x,y} \right) > 0} \right\}\)is a rectangular region R (possibly unbounded) with sides (if any) parallel to the coordinate axes. Then X and Y are independent if and only if Eq. (3.5.7) holds for all\(\left( {x,y} \right) \in R\)
Suppose that a random variableXhas a discrete distribution
with the following p.f.:
\(f\left( x \right) = \left\{ \begin{array}{l}cx\;\;for\;x = 1,...,5,\\0\;\;\;\;otherwise\end{array} \right.\)
Determine the value of the constantc.
Suppose that thenrandom variablesX1. . . , Xnform arandom sample from a discrete distribution for which thep.f. is f. Determine the value of Pr(X1 = X2 = . . .= Xn).
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) \({\rm P}\left( {Y > 0|x = 2} \right)\)
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