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Chapter 1: Problem 6

Let \(X\) have the pdf \(f(x)=3 x^{2}, 0

Short Answer

Expert verified
The expected value of the area of the rectangle is 0.15

Step by step solution

01

Write down the formula for the area of the rectangle

The area of the rectangle, \(A\), is the product of its sides, \(X\) and \((1-X)\). So, \(A = X(1-X) = X - X^2\). This is the function for which we will find the expected value.
02

Write down the formula for the expected value of a function of a random variable

The expected value of a function of a random variable, \(g(X)\), with respect to a pdf \(f(x)\) is given by \(\mathbb{E}(g(X)) = \int g(x)f(x) dx\). We will use this equation with \(g(X) = X - X^2\) and \(f(x) = 3x^2\).
03

Compute the integral

Substitute \(g(X)\) and \(f(x)\) into the equation for \(\mathbb{E}(g(X))\): \(\mathbb{E}(X - X^2) = \int_{0}^{1} (X - X^2)3x^2 dx = 3\int_{0}^{1} (x^3 - x^4) dx\). This integral can be solved easily: \(\int x^3 dx = 0.25x^4 |_{0}^{1} = 0.25\) and \(\int x^4 dx = 0.2x^5 |_{0}^{1} = 0.2\). Therefore, \(\mathbb{E}(X - X^2) = 3 (0.25 - 0.2) = 0.15\).

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Most popular questions from this chapter

To join a certain club, a person must be either a statistician or a mathematician or both. Of the 25 members in this club, 19 are statisticians and 16 are mathematicians. How many persons in the club are both a statistician and a mathematician?

Let a random variable \(X\) of the continuous type have a pdf \(f(x)\) whose graph is symmetric with respect to \(x=c .\) If the mean value of \(X\) exists, show that \(E(X)=c\) Hint: Show that \(E(X-c)\) equals zero by writing \(E(X-c)\) as the sum of two integrals: one from \(-\infty\) to \(c\) and the other from \(c\) to \(\infty\). In the first, let \(y=c-x\); and, in the second, \(z=x-c\). Finally, use the symmetry condition \(f(c-y)=f(c+y)\) in the first.

If a sequence of sets \(C_{1}, C_{2}, C_{3}, \ldots\) is such that \(C_{k} \subset C_{k+1}, k=1,2,3, \ldots\), the sequence is said to be a nondecreasing sequence. Give an example of this kind of sequence of sets.

Let \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the \(\mathrm{mgf}\) is $$ M(t)=\left\\{\begin{array}{ll} \frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0 \end{array}\right. $$

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