Problem 122 - An Introduction to Mathematical Statistics and Its Applications

Question

. Suppose \(U\) is a uniform random variable over \([0,1]\). (a) Show that \(Y=(b-a) U+a\) is uniform over \([a, b]\). (b) Use part (a) and Theorem \(3.6 .2\) to find the variance of \(Y\).

Step-by-Step Solution

Answer

In part (a), \(Y\) is demonstrated to be a uniform random variable over the interval \([a, b]\). In part (b), the variance of \(Y\) is computed as \((b - a)^{2}/12\) according to Theorem \(3.6.2\).

Step 1: Part (a) Proof of Uniformity

Uniform random variable \(U\) over \([0,1]\) has the probability density function (PDF) \(f_U(u) = 1\) for \(0 \leq u \leq 1\). Then, for \(Y=(b-a)U+a\), we can find the PDF of \(Y\). By substituting \(U\) as \((Y-a)/(b-a)\) in \(f_U(u)\), we obtain \(f_Y(y) = 1/(b-a)\) for \(a \leq y \leq b\). This is the PDF of a uniform distribution over \([a, b]\). Therefore, \(Y\) is a uniform random variable over \([a, b]\).

Step 2: Part (b) Calculation of Variance

The variance of a uniform random variable \(Y\) over the interval \([a, b]\) can be found using Theorem \(3.6.2\). According to the theorem, the variance \(\text{Var}(Y)\) equals \((b - a)^{2}/12\). Hence, applying the theorem to the present problem provides the variance of \(Y\).

Key Concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in statistics that describes the likelihood of a random variable taking on particular values. Specifically, the PDF of a continuous random variable, like our uniform random variable U, gives the probability that the variable falls within a certain range. It is defined such that the area under the curve of the PDF over a range of values yields the probability that the random variable falls within that range. For a uniform distribution over the interval [0,1], the PDF is particularly simple: it is a constant value as long as we are within the interval and zero otherwise.

This is expressed mathematically for U as:
\[ f_U(u) = 1 \text{ for } 0 \leq u \leq 1, \]and zero elsewhere. This rectangular shape of the PDF reflects the 'uniform' nature of the distribution: every outcome in the interval is equally likely. When we transform U to Y, through a linear transformation, the range changes but the uniformity remains intact, which is crucial for understanding the behavior of Y.

Variance of a Random Variable

The variance of a random variable is a measure of how much the values of the variable are spread out from the mean. It quantifies the dispersion of the distribution and is always non-negative. A higher variance indicates that the data points are more spread out. In the context of a uniform random variable Y, variance tells us how the outcomes within the interval [a,b] are distributed around their average value.

For any uniform distribution over an interval [a, b], the variance is given by the formula: \[ \text{Var}(Y) = \frac{(b - a)^2}{12} \].
This formula comes from the general definition of variance, which is the expected value of the squared deviation from the mean. The result (b - a)^2/12 simplifies the computation and directly ties the spread of a uniform distribution to the length of the interval over which it's defined. The factor of 12 in the denominator shows a uniform distribution's intrinsic properties, which are constant regardless of the particular interval chosen.

Uniform Distribution Properties

The uniform distribution is one of the simplest probability distributions and has several key properties that make it unique. Firstly, because any interval of the same length within the distribution's range has an equal probability, the distribution is said to be 'uniform'. This property stands in contrast to other distributions, like the normal distribution, where the probabilities are not evenly distributed.

Some important properties of the uniform distribution are:

• Continuity: The uniform distribution is continuous, meaning that it takes on an infinite number of values within a certain range.
• Constant PDF: Within its bounds, the PDF of a uniform distribution is constant, reflecting equal likelihood of all outcomes within that range.
• Simple Mean and Variance: For a uniform distribution over [a, b], the mean is (a + b)/2, and the variance is (b - a)^2/12, as shown in the exercise.
• Symmetry: The uniform distribution is symmetric about its mean.

The understanding of these properties helps students to grasp the concept of uniformity in a distribution and the implications it has on the behavior of random variables that adhere to it.