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Chapter 10: Problem 66

Assume that the random variable \(X\) is normally distributed. Use the given information to find the unknown parameter or parameters of the distribution. If \(S D(X)=3\) and \(P(X \geq 2)=0.6293\), find \(E(X)\)

Short Answer

Expert verified
E(X) = 2.9

Step by step solution

01

Understand the Problem

The problem provides that the random variable X is normally distributed with a standard deviation, which is denoted here as SD(X) = 3. Additionally, it provides a probability statement regarding X: P(X 鈮� 2) = 0.6293. The goal is to find the expected value of X, denoted as E(X).
02

Convert Probability Statement to Z-Score

Since we are dealing with a normal distribution, convert the given probability into a z-score. Recall that the Z-score formula for a normal distribution is given by \[ Z = \frac{X - \mu}{\sigma} \] where \mu is the mean (E(X)) and \sigma is the standard deviation (SD(X)). Using the Z-table (or a standard normal distribution table), find the Z-score that corresponds to P(Z 鈮� z) = 0.6293.
03

Use Z-Score to Compute Z Value

To find the z-score that corresponds to P(Z 鈮� z) = 0.6293, we first note that P(Z 鈮� z) = 1 - 0.6293 = 0.3707. Using the Z-table, a probability of 0.3707 corresponds to a z-score of approximately -0.3.
04

Set Up the Z-Score Equation

Use the z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] Since X = 2, \sigma = 3, and Z = -0.3, substitute these values into the equation: \[ -0.3 = \frac{2 - \mu}{3} \]
05

Solve for \mu (E(X))

Rearrange the equation to solve for E(X): \[ -0.3 = \frac{2 - E(X)}{3} \] Multiply both sides by 3 to isolate the term involving \mu: \[ -0.3 \times 3 = 2 - E(X) \] \[ -0.9 = 2 - E(X) \] Add E(X) and 0.9 to both sides: \[ 2 + 0.9 = E(X) \] Therefore, \[ E(X) = 2.9 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution
The normal distribution is a fundamental concept in statistics. It describes a continuous probability distribution where most of the observations cluster around the central peak. This peak is the mean or average of the data.
Key properties of the normal distribution include:
  • Symmetry around the mean
  • Mean, median, and mode are equal
  • Defined by its mean (u) and standard deviation (u)
In this exercise, you're dealing with a normally distributed random variable, which means the data follows this bell-shaped curve.
Standard Deviation
Standard deviation is a measure of the amount of variability or spread in a set of data. In simpler terms, it tells us how much the values in a data set deviate from the mean.
To calculate the standard deviation (u), the formula is:
\[u = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (X_i \space - \space \mu)^2}\] where:
  • u is the standard deviation
  • N is the number of observations
  • Xi represents each data point
  • u is the mean of the data
In the given problem, the standard deviation (u) is provided as 3.
Expected Value
The expected value (u) represents the mean or average value of a random variable in probability and statistics. It is a measure of the central tendency.
Mathematically, for a discrete random variable, the expected value is represented as:
\[u(X) = \sum_{i=1}^{N} X_i \space P(X_i)\] where:
  • u(X) is the expected value
  • Xi represents each possible value of the random variable
  • P(Xi) is the probability of Xi occurring
In this exercise, you are required to find E(X), which represents the expected value or mean of the normal distribution when you know the standard deviation and a probability related to the random variable.
Z-Score
The Z-Score, also known as the standard score, measures how many standard deviations an element is from the mean. It helps in determining the position of a specific value within a dataset in relation to the mean.
The Z-Score formula is:
\[u = \frac{X \space - \space u}{u}\] where:
  • Z is the Z-score
  • X is the value in question
  • u is the mean
  • u is the standard deviation
In the provided exercise, to find the unknown mean (u), you start by converting the given probability to a Z-score using the Z-table. Once the Z-score is identified, the remaining values are substituted into the formula to solve for u (expected value).

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Most popular questions from this chapter

A significance level \(\alpha\) and a tail of the standard normal distribution are given. Use the normal table to approximately determine the critical value. \(\alpha=0.05\), left tail

Suppose \(X\) has the Binomial \((n, p)\) distribution. Use the normal approximation to estimate the given probability. \(P(X \geq 18)\) if \(n=100, p=0.2\)

Let \(Z\) be a Normal \((0,1)\) random variable. Find the probability that \(Z\) is in the interval. $$ [-0.26,0.7] $$

Suppose that an individual is randomly selected from a population that satisfies the Hardy-Weinberg Law. Let \(p\) and \(q\) denote the proportions of \(A\) and \(a\) alleles in the population. a) The allele inherited by the offspring may be visualized with the following multiplication tree. Fill in the probabilities of the unmarked arrows. b) In the multiplication tree, there are three ways corresponding to the transmission of the \(A\) allele. Show that the probability that the \(A\) allele is transmitted is \(p\). c) Show that the probability that the \(a\) allele is Iransmitted is \(q .\) d) Use the result of Example 9 to show that, if one generation satisfies the Hardy-Weinberg Law and survives to mate, then the offspring generation should also satisfy it.

Compute the probability using the given distribution. \(P(0 \leq X \leq 3), X\) is Uniform \((0,7)\)
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