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Chapter 2: Problem 57

\(X \sim N\left(\mu, \sigma^{2}\right)\) and \(Y=a X+b,\) where \(a<0,\) show that \(Y \sim N\left(a \mu+b, a^{2} \sigma^{2}\right).\)

Short Answer

Expert verified
Y follows a normal distribution: \(Y \sim N(a\mu + b, a^2\sigma^2)\).

Step by step solution

01

Understand the Transformation

We are given the random variable \(X\) which follows a normal distribution with mean \(\mu\) and variance \(\sigma^2\), i.e., \(X \sim N(\mu, \sigma^2)\). The transformed variable \(Y\) is defined as \(Y = aX + b\), where \(a < 0\). Our goal is to find out if \(Y\) also follows a normal distribution and to identify its parameters.
02

Identify the Mean of Y

For a linear transformation \(Y = aX + b\), the expected value (mean) of \(Y\) can be found using the formula for the mean of a linear transformation: \(E(Y) = aE(X) + b\). Therefore, the mean of \(Y\) is obtained as follows:\[E(Y) = a\mu + b\]
03

Determine the Variance of Y

The variance of a linear transformation \(Y = aX + b\) is given by the formula \(\text{Var}(Y) = a^2 \text{Var}(X)\). Since \(X\) has variance \(\sigma^2\), the variance of \(Y\) is:\[\text{Var}(Y) = a^2\sigma^2\]
04

Conclude the Distribution of Y

In a linear transformation of a normally distributed variable, the result is also normally distributed. Thus, \(Y\) is normally distributed with parameters derived in previous steps: mean \(a\mu + b\) and variance \(a^2\sigma^2\). Therefore, the distribution of \(Y\) is:\[Y \sim N(a\mu + b, a^2\sigma^2)\]
05

Conclusion and Verification

The transformation preserves the normality due to its linear nature. Even though \(a < 0\), this does not affect the normality but merely changes the sign of the slope, confirming that the derived parameters are correct.

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Key Concepts

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

Linear Transformation
A linear transformation occurs when a random variable undergoes a specific kind of modification. In the context of normal distribution, it involves conversion using a formula of the form: \(Y = aX + b\). The variables \(a\) and \(b\) are constants, and in this exercise, \(a < 0\). This signifies a type of scaling and shifting of the distribution.

  • Scaling alters the spread of the data. A negative \(a\) flips the distribution around, reversing its direction.
  • Shifting, through \(b\), simply moves the entire distribution along the axis, either up or down.
Despite these changes, a linear transformation of a random variable with a normal distribution will yield another normal distribution. This resilience highlights the robustness of the normal distribution under linear operations.
Mean and Variance
Mean and variance are fundamental to understanding the shape and spread of a distribution. For any random variable \(X\), they help us know where the data is centered and how widely it is dispersed.

  • The mean \(\mu\) represents the average or central tendency of the data. For a transformed variable \(Y = aX + b\), its mean is \(E(Y) = a\mu + b\). This formula shows how the changes in \(a\) and \(b\) affect the new mean.
  • Variance, denoted by \(\sigma^2\), tells us how much the values are spread out. For \(Y\), its variance is \(\text{Var}(Y) = a^2\sigma^2\). The squaring of \(a\) ensures the variance is always positive, maintaining the properties of the normal distribution.
These calculations ensure that even when the distribution is altered, the key statistical properties鈥攎ean and variance鈥攁re readily and accurately evaluated.
Random Variable
A random variable is a concept that involves the assignment of numeric values to each outcome in a probabilistic experiment. In this exercise, the random variable \(X\) follows a normal distribution with a particular mean and variance.

  • This distribution is described famously by the bell curve, where the shape is symmetrical around the mean.
  • The property \(X \sim N(\mu, \sigma^2)\) illustrates this distribution, characterized by its mean and variance.
When we apply a transformation to create a new random variable, \(Y = aX + b\), the nature of the normal distribution is preserved.

  • Transformations do not disrupt the underlying structure of the normal distribution.
  • Linear transformations map quantiles from one distribution to another seamlessly, maintaining normality.
Understanding how random variables behave under transformation is crucial for statistical reasoning and analysis, revealing how flexible and adaptable distributions are, even under significant alterations.

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Most popular questions from this chapter

A multiple-choice test consists of 20 items, each with four choices. A student is able to eliminate one of the choices on each question as incorrect and chooses randomly from the remaining three choices. A passing grade is 12 items or more correct. a. What is the probability that the student passes? b. Answer the question in part (a) again, assuming that the student can eliminate two of the choices on each question.

Find the cdf of a Bernoulli random variable.

An experiment consists of throwing a fair coin four times. Find the frequency function and the cumulative distribution function of the following random variables: (a) the number of heads before the first tail, (b) the number of heads following the first tail, (c) the number of heads minus the number of tails, and (d) the number of tails times the number of heads.

For what value of \(k\) is the Poisson frequency function with parameter \(\lambda\) maximized? (Hint: Consider the ratio of consecutive terms.)

If \(X\) is a geometric random variable, show that $$P(X>n+k-1 | X>n-1)=P(X>k)$$ In light of the construction of a geometric distribution from a sequence of independent Bernoulli trials, how can this be interpreted so that it is "obvious"?
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