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Chapter 11: Problem 50

If a continuous random variable \(X\) has mean \(5,\) then what is the mean of \(X+100 ?\) Of \(2 X ?\)

Short Answer

Expert verified
Mean of \(X+100\) is 105; mean of \(2X\) is 10.

Step by step solution

01

Understanding the Mean of Transformed Variables

If a random variable \( X \) has a mean \( \mu_X \), then the mean of the random variable \( X + c \) (where \( c \) is a constant) is given by \( \mu_{X+c} = \mu_X + c \). Similarly, the mean of the random variable \( aX \) (where \( a \) is a constant) is \( \mu_{aX} = a \cdot \mu_X \). These transformations follow from the properties of expected values in probability theory.
02

Calculate the Mean of X + 100

Using the property \( \mu_{X+c} = \mu_X + c \), where \( \mu_X = 5 \) and \( c = 100 \), we find the mean of \( X + 100 \) as follows: \[ \mu_{X+100} = 5 + 100 = 105. \]
03

Calculate the Mean of 2X

Using the property \( \mu_{aX} = a \cdot \mu_X \), where \( \mu_X = 5 \) and \( a = 2 \), we determine the mean of \( 2X \) by calculating: \[ \mu_{2X} = 2 \cdot 5 = 10. \]

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Key Concepts

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

Understanding the Mean of Transformed Variables
When working with a random variable, transforming it can be crucial for various applications in statistics and probability. The mean or expected value provides a central measure for the random variable. However, what happens if we transform the variable? Various rules apply:
  • If you add a constant \( c \) to the random variable \( X \), the mean of this new variable, \( X + c \), will simply be the mean of \( X \) plus the constant \( c \). Mathematically, denoted as \( \mu_{X+c} = \mu_X + c \).
  • If you multiply the random variable \( X \) by a constant \( a \), the mean of this new variable, \( aX \), will be \( a \) times the mean of \( X \). Mathematically, this is written as \( \mu_{aX} = a \cdot \mu_X \).
These transformations offer powerful insights especially when modeling real-world phenomena. It makes the manipulation of data straightforward and intuitive, sticking to basic arithmetic operations to understand new variable means.
Exploring Expected Value Properties
Expected value, or mean, holds several unique properties that simplify mathematical computations. The expected value is essentially a weighted average of all possible values a random variable can take, weighted by their probabilities.
  • Linearity of Expectation: One of the core properties is that expected value is linear. This means \( E[X + Y] = E[X] + E[Y] \) no matter if \( X \) and \( Y \) are dependent or independent.
  • Expected Value of a Constant: Another property is that the expected value of a constant is just the constant itself. Therefore, \( E[c] = c \).
  • Multiplication by a Scalar: For any constant \( a \), \( E[aX] = a \cdot E[X] \). This property is crucial for scaling variables.
These properties promote simplification and demystification of complex probabilistic scenarios, proving to be invaluable tools in both theoretical and applied mathematics.
Basics of Continuous Random Variables
Continuous random variables are those that can take an infinite number of values within a given range. Unlike discrete random variables, which have specific values (like integers), continuous ones deal with ranges of values.
  • A continuous random variable is often represented by a probability density function (PDF), which describes the likelihood of the variable taking on a specific value.
  • The area under the PDF curve represents probability, with the total area equating to 1.
  • Mean or expected value of a continuous random variable is computed using the integral of the product of the variable and its probability density function over its entire range.
Understanding continuous random variables is essential because they are widely used in fields like physics, finance, and engineering. They help in modeling various real-world continuous phenomena like time, temperature, and distance, making them a cornerstone of statistical analysis.

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Most popular questions from this chapter

For a person who has received treatment for a life-threatening disease, such as cancer, the number of years of life after the treatment (the survival time) can be modeled by an exponential random variable. The survival function \(S(x)\) gives the probability that a randomly selected patient will survive for at least \(x\) years after treatment has ended. Show that \(S(x)=1-F(x),\) where \(F(x)\) is the cumulative distribution function for the exponential random variable, and find a formula for \(S(x)\) in terms of the exponential parameter \(a\).

Find each probability for a standard normal random variable \(Z\). \(P(Z<-3.1)\)

Find each probability for a standard normal random variable \(Z\). \(P(-1.5 \leq Z \leq 1.05)\)

Modern portfolio theory generally assumes that the yield on a diversified portfolio of investments will be normally distributed around its historic mean. What is the probability that a diversified portfolio will yield at least \(5 \%\) if the historic mean is \(8 \%\) and standard deviation \(1.5 \% ?\)

If \(X\) has probability density function \(f(x)\), then what is the probability density function for \(X+5 ?\) [Hint: Think of how the graphis shifted.]
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