Question

Let the random variable \(X\) have a distribution that is \(N\left(\mu, \sigma^{2}\right)\). (a) Does the random variable \(Y=X^{2}\) also have a normal distribution? (b) Would the random variable \(Y=a X+b, a\) and \(b\) nonzero constants have a normal distribution? Hint: \(\quad\) In each case, first determine \(P(Y \leq y)\).

Short answer

(a) No, the variable \(Y=X^2\) does not have a normal distribution. (b) Yes, the random variable \(Y=a X+b\) would have a normal distribution.

Step-by-step solution

(a) Analysis and Solution

We first address if the variable \(Y = X^2\) would be normally distributed. Note that, a random variable \(X\) is normally distributed if it has a symmetric bell-shaped density curve. But the square of a normally distributed variable \(X\) will have a distribution that's not symmetric and is confined to positive values, meaning it cannot have negative values. This denotes that \(Y = X^2\) does not follow a normal distribution.

(b) Analysis and Solution

Next, we look at \(Y = aX + b\), where \(a\) and \(b\) are non-zero constants. Y is a linear transformation of X. Linear transformations of a normal random variable always results in another normal random variable. Therefore, \(Y = aX + b\) would be normally distributed.

Key concepts

Random Variable

In the world of statistics and probability, a random variable plays a pivotal role. It is a numerical description of the outcome of a statistical experiment, essentially a variable whose value is unknown until the experiment is performed. The random variable is denoted by a letter, such as X in our exercise, which is used to describe events that are not deterministic but have a certain distribution of probabilities. For example, if we roll a six-sided die, the outcome is a random variable that can take on any value between 1 and 6 with equal probability.

Random variables are classified into two main types: discrete and continuous. Discrete random variables have a countable number of possible values. In contrast, continuous random variables, much like our variable X in the exercise, can take on any value in a given range and are described by a probability density function. The normal distribution, also known as Gaussian distribution, is a fundamental probability distribution for a continuous random variable, characterized by its mean (mu, μ) and variance (sigma squared, σ²), which determine its shape and spread.

Linear Transformation

A linear transformation is a powerful concept in statistics wherein a new random variable is created from an existing one through a specific operation. This operation involves multiplying the original variable by a constant and then adding another constant. Mathematically, if we have a random variable X, and constants a and b, we can define a new random variable Y as Y = aX + b.

The beauty of a linear transformation when applied to a random variable with a normal distribution is that it preserves normality. This means that if X is normally distributed, then Y, defined as a linear transformation of X, will also have a normal distribution. It is due to this property that we can assert that Y = aX + b (with a and b being non-zero constants) in the given exercise will indeed have a normal distribution.

Understanding linear transformations is essential in statistics, as it allows us to scale and shift data while maintaining the inherent properties of the original distribution, which can be very useful for data analysis and interpretation.

Probability

The concept of probability is fundamental to understanding random variables and their distributions. Probability is a measure of the likelihood that a particular event will occur. It quantifies uncertainty and is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 signifies certainty.

In statistics, we often use probability to describe the distribution of a random variable. For example, when we say a random variable X follows a normal distribution N(μ, σ²), we are expressing the probability of X taking on any given value within that distribution. Calculating the probability that a random variable will be less than or equal to a certain value, expressed as P(X ≤ x), is key to making predictions and providing insights into the behavior of the variable.

In the context of our exercise, understanding probability allows us to determine whether transformations of a normally distributed variable will result in distributions that continue to predict the likelihood of events occurring in a similar manner. This is crucial when studying scenarios that follow natural or near-natural patterns, as many real-world phenomena can be approximately modeled using normal distributions.