/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Let \(X_{1}, X_{2}, \ldots, X_{n... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 37

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables, each having a uniform distribution over \((0,1)\). Let \(M=\) maximum \(\left(X_{1}, X_{2}, \ldots, X_{n}\right)\). Show that the distribution function of \(M, F_{M}(\cdot)\), is given by $$ F_{M}(x)=x^{n}, \quad 0 \leq x \leq 1 $$ What is the probability density function of \(M ?\)

Short Answer

Expert verified
The probability density function of \(M\), the maximum of \(n\) independent random variables with a uniform distribution over (0,1), is given by: $$ f_M(x) = n \cdot x^{(n-1)}, \quad 0 \le x \le 1 $$

Step by step solution

01

Find the probability of \(M \le x\)

We want to find the probability that the maximum of the random variables \(X_1, X_2, \dots, X_n\) is less than or equal to \(x\), i.e., \(P(M \le x)\). Since the random variables are independent, we can find the product of their individual probabilities of being less than or equal to \(x\): $$ P(M \le x) = P(X_1 \le x) \cdot P(X_2 \le x) \cdots P(X_n \le x) $$
02

Compute individual probabilities

Since each random variable \(X_i\) has a uniform distribution over (0,1), their cumulative distribution functions (CDFs) can be written as: $$ P(X_i \le x) = \begin{cases} 0 & x < 0 \\ x & 0 \le x \le 1 \\ 1 & x > 1 \end{cases} $$
03

Compute the cumulative distribution function of \(M\), \(F_M(x)\)

Using the individual probabilities, we can now find \(F_M(x) = P(M \le x)\). Since the range of the variables is (0,1), we can ignore the cases when \(x < 0\) and \(x > 1\): $$ F_M(x) = P(M \le x) = P(X_1 \le x) \cdot P(X_2 \le x) \cdots P(X_n \le x) = x \cdot x \cdots x = x^n $$ So, the cumulative distribution function of \(M\) is \(F_M(x) = x^n\) for \(0 \le x \le 1\).
04

Find the probability density function of \(M\)

To find the probability density function (PDF) of \(M\), we need to differentiate the cumulative distribution function with respect to \(x\): $$ \frac{d}{dx} F_M(x) = \frac{d}{dx} x^n = n \cdot x^{(n-1)} $$ Thus, the probability density function of \(M\) is: $$ f_M(x) = n \cdot x^{(n-1)}, \quad 0 \le x \le 1 $$

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Most popular questions from this chapter

Suppose that each coupon obtained is, independent of what has been previously obtained, equally likely to be any of \(m\) different types. Find the expected number of coupons one needs to obtain in order to have at least one of each type. Hint: Let \(X\) be the number needed. It is useful to represent \(X\) by $$ X=\sum_{i=1}^{m} X_{i} $$ where each \(X_{1}\) is a geometric random variable.

Suppose that \(X\) and \(Y\) are independent continuous random variables. Show that $$ P\left[X \leq Y \mid=\int_{-\infty}^{\infty} F_{x}(y) f_{Y}(y) d y\right. $$

Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time \(n\) if \(X_{n}>\max \left(X_{1}, \ldots, X_{n-1}\right) .\) That is, \(X_{n}\) is a record if it is larger than each of \(X_{1+\ldots,} X_{n-1} .\) Show (i) \(P\) a record occurs at time \(n]=1 / n\) (ii) \(E\) [number of records by time \(n]=\sum_{i=1}^{n} 1 / i\) (iii) Var(number of records by time \(n)=\sum_{i=1}^{n}(i-1) / i^{2}\) (iv) Let \(N=\min [n: n>1\) and a record occurs at time \(n] .\) Show \(E[N]=\infty\). Hint: For (ii) and (iii) represent the number of records as the sum of indicator (that is, Bernoulli) random variables.

For the multinomial distribution (Exercise 17\()\), let \(N_{t}\) denote the number of times outcome \(i\) occurs. Find (i) \(E\left[N_{j}\right]\) (ii) \(\operatorname{Var}\left(N_{i}\right)\) (iii) \(\operatorname{Cov}\left(N_{i}, N_{j}\right)\) (iv) Compute the expected number of outcomes which do not occur.

An urn contains five red, three orange, and two bluc balls. Two balls are randomly selected. What is the sample space of this experiment? Let \(X\) represent the number of orange balls selected. What are the possible values of \(X\) ? Calculate \(P[X=0\\}\).
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