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Chapter 4: Q 5SE (page 272)
Suppose that the random variable X has to mean\(\mu \), and variance\({\sigma ^2}\)and that\(Y = aX + b\)Determine the values of a and b for which\(E\left( Y \right) = 0\)and\(Var\left( Y \right) = 1\)
The value of a is\( \pm \frac{1}{{{\sigma ^2}}}\)
The value of b is \( - au\)
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Let X have the discrete uniform distribution on the integers \(1, \ldots \ldots ,n\).Compute the variance of X.
Hint: Youmay wish to use the formula \(\sum\limits_{k = 1}^n {{k^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)
Suppose that one word is to be selected at random from the sentence the girl put on her beautiful red hat. If X denotes the number of letters in the word that is selected, what is the value of E(X)?
Suppose thatXis a random variable for which the m.g.f. is as follows:\(\psi \left( t \right) = \frac{1}{6}\left( {4 + {e^t} + {e^{ - t}}} \right)\)for鈭掆垶< t <鈭�. Find the probability distribution ofX.
Suppose thatXhas the uniform distribution on the interval (a, b). Determine the m.g.f. ofX.
If n houses are located at various points along a straight road, at what point along the road should a store be located in order to minimize the sum of the distances from the n houses to the store?
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