/*! 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 6 Let \(X_{1}, X_{2}, \ldots, X_{n... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 6

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\mu, \sigma^{2}\right)\). (a) If the constant \(b\) is defined by the equation \(P(X \leq b)=0.90\), find the mle of \(b\). (b) If \(c\) is given constant, find the mle of \(P(X \leq c)\).

Short Answer

Expert verified
The MLE of \( b \) is given as \( b = \mu + 1.282*\sigma \), and the MLE of \( P(X \leq c) \) is given as \( 桅\left(\frac{c - \mu}{\sigma}\right) \).

Step by step solution

01

Identify the Maximum Likelihood Estimation (MLE) of b

For a normal distribution \( N\left(\mu,\sigma^{2}\right) \), the probability \( P(X \leq b)=0.90 \) refers to the 90th percentile (quantile). In terms of the z-score (the number of standard deviations from the mean), this corresponds to a Z-value of approximately 1.282. In the standard normal distribution table, this Z-score corresponds with the given 0.90 probability. So, the maximum likelihood estimation of \( b \) will approximately equivalent to \( \mu + 1.282*\sigma \).
02

Identify the Maximum Likelihood Estimation (MLE) of P(X 鈮� c)

In the second part, \( c \) is a given constant. The MLE of \( P(X \leq c) \) is the cumulative distribution function (CDF) of \( X \) evaluated at \( c \). For a normal distribution \( N\left(\mu,\sigma^{2}\right) \), this can be denoted as \( 桅\left(\frac{c - \mu}{\sigma}\right) \), where \( 桅(x) \) stands for the standard normal cumulative distribution function.

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Most popular questions from this chapter

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