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Chapter 4: Q4.1-2E (page 216)

If an integer between 1 and 100 is to be chosen at random, what is the expected value?

Short Answer

Expert verified

The expected value of the integer is 45.5

Step by step solution

01

Given information

An integer is chosen randomly, and we have the expected value of the chosen integer.

02

Compute the expected value

Let X be the random variable that a number is chosen at random. Here, X has the uniform distribution over the interval \(\left[ {1,100} \right]\) where the probability mass function f(x) is given by

\(\begin{array}{l}f(x) = \frac{1}{{100}},\,1 \le x \le 100\\\,\,\,\,\,\,\,\,\,\,\,\, = 0,\,{\rm{otherwise}}\end{array}\)

The expected value of X denoted by \(E(X)\) is given by

\(\begin{array}{l}E(X) = \sum\limits_x {xf(x)} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sum\limits_x {x\,\frac{1}{{100}}} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{{100}} \times \frac{{100(100 - 1)}}{2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 45.5\end{array}\) [ The sum of first n natural numbers is \(\frac{{n(n - 1)}}{2}\) ]

So, the expected value of the integer is 45.5

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Most popular questions from this chapter

Suppose that a random variable X has a continuous distribution with the p.d.f. has given in Example 4.1.6. Find the expectation of 1/X

Suppose that\({\bf{X}}\)and\({\bf{Y}}\)are random variables such that

\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = 9}}\),\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ = 4}}\),and\({\bf{\rho }}\left( {{\bf{X,Y}}} \right){\bf{ = - }}\frac{{\bf{1}}}{{\bf{6}}}\).Determine

(a)\({\bf{Var}}\left( {{\bf{X + Y}}} \right)\)and(b)\({\bf{Var}}\left( {{\bf{X - 3Y + 4}}} \right)\).

Let X be a random variable with c.d.f F. Suppose that\(a < b\)are numbers such that both a and b are medians of X.

  1. Prove that\(F\left( a \right) = \frac{1}{2}\).
  2. Prove that there exists a smallest \(c \le a\)and a largest\(d \ge b\)such that every number in the closed interval\(\left( {c,d} \right)\) is a median of X.
  3. If X has a discrete distribution, prove that \(F\left( d \right) > \frac{1}{2}\).

Suppose that the random variables X1, . . . , Xnform a random sample of sizenfrom a continuous distribution for which the c.d.f. isF, and let the random variables \({{\bf{Y}}_{\bf{1}}}\) and \({{\bf{Y}}_{\bf{n}}}\)be defined as in Exercise 11. Find \({\bf{E[F(}}{{\bf{Y}}_{\bf{1}}}{\bf{)]}}\)and \({\bf{E[F(}}{{\bf{Y}}_{\bf{n}}}{\bf{)]}}\)Let \({{\bf{Y}}_{\bf{1}}}{\bf{ = min\{ }}{{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}...{{\bf{X}}_{\bf{n}}}{\bf{\} }}\), and let \({{\bf{Y}}_{\bf{n}}}{\bf{ = max\{ }}{{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}...{{\bf{X}}_{\bf{n}}}{\bf{\} }}\)

Suppose that X, Y , and Z are nonnegative random variables such that\({\rm P}\left( {X + Y + Z < 1.3} \right) = 1\). Show that X, Y , and Z cannot possibly have a joint distribution under which each of their marginal distributions is the uniform distribution on the interval\(\left( {0,1} \right)\).
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