Chapter 3: Q9E (page 107)
Show that there does not exist any numbercsuch that the following functionf (x)would be a p.d.f.:

Short Answer
There does not exist any number c such that

is a probability density function.
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Chapter 3: Q9E (page 107)
Show that there does not exist any numbercsuch that the following functionf (x)would be a p.d.f.:

There does not exist any number c such that

is a probability density function.
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Suppose that a random variableXhas the binomial distribution
with parametersn=8 andp=0.7. Find Pr(X≥5)by using the table given at the end of this book. Hint: Use the fact that Pr(X≥5)=Pr(Y≤3), whereYhas thebinomial distribution with parametersn=8 andp=0.3.
Suppose that an electronic system comprises four components, and let\({X_j}\)denote the time until component j fails to operate (j = 1, 2, 3, 4). Suppose that\({X_1},{X_2},{X_3}\)and\({X_4}\)are i.i.d. random variables, each of which has a continuous distribution with c.d.f.\(F\left( x \right)\)Suppose that the system will operate as long as both component 1 and at least one of the other three components operate. Determine the c.d.f. of the time until the system fails to operate.
Suppose that a box contains seven red balls and three blue balls. If five balls are selected at random, without replacement, determine the p.f. of the number of red balls that will be obtained.
Suppose that the p.d.f. of X is as given in Exercise 3. Determine the p.d.f. of\(Y = 4 - {X^3}\)
Suppose that the joint distribution of X and Y is uniform over the region in the\({\bf{xy}}\)plane bounded by the four lines\({\bf{x = - 1,x = 1,y = x + 1}}\)and\({\bf{y = x - 1}}\). Determine (a)\({\bf{Pr}}\left( {{\bf{XY > 0}}} \right)\)and (b) the conditional p.d.f. of Y given that\({\bf{X = x}}\).
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