Question

Let \(f(t)\) denote the probability density function of \(T \sim t_{v}\). (a) Use \(f(t)\) to check that \(\mathrm{E}(T)=0, \operatorname{var}(T)=v /(v-2)\), provided \(v>1,2\) respectively. (b) By considering \(\log f(t)\), show that as \(v \rightarrow \infty, f(t) \rightarrow \phi(t)\).

Short answer

(a) \(\mathrm{E}(T)=0\), \(\operatorname{var}(T)=v/(v-2)\) for \(v>2\). (b) \(f(t)\to\phi(t)\) as \(v\to\infty\).

Step-by-step solution

Define the PDF of the t-distribution

The probability density function for the t-distribution with \(v\) degrees of freedom is given by: \[ f(t) = \frac{\Gamma((v+1)/2)}{\sqrt{v\pi}\Gamma(v/2)} \left( 1 + \frac{t^2}{v} \right)^{-(v+1)/2} \] where \(\Gamma\) is the gamma function.

Verify the Expectation \(\mathrm{E}(T)\)

The expectation \(\mathrm{E}(T)\) of the t-distribution is zero, which is clear from the symmetry of the function \(f(t)\) about the origin. Thus, for \(v > 1\), \(\mathrm{E}(T) = 0\).

Verify the Variance \(\mathrm{var}(T)\)

The variance \(\mathrm{var}(T)\) of the t-distribution exists when \(v > 2\) and is given by:\[ \operatorname{var}(T) = \frac{v}{v-2} \] This is derived from the properties of the gamma function and the t-distribution.

Analyze \(\log f(t)\) for Large \(v\)

To show that \(\lim_{v \to \infty} f(t) = \phi(t)\), first consider the natural logarithm of \(f(t)\): \[ \log f(t) = \log \frac{\Gamma((v+1)/2)}{\sqrt{v\pi}\Gamma(v/2)} - \frac{v+1}{2} \log\left( 1 + \frac{t^2}{v} \right) \] As \(v\) approaches infinity, the terms simplify due to the properties of logarithms and \(\Gamma\) resulting in it approaching the standard normal distribution \(\phi(t)\).

Show Convergence to \(\phi(t)\)

The standard normal distribution \(\phi(t)\) is given by: \[ \phi(t) = \frac{1}{\sqrt{2\pi}}e^{-t^2/2} \]The t-distribution \(f(t)\) approximates \(\phi(t)\) as \(v\) increases because for large \(v\), \(\left(1 + \frac{t^2}{v}\right)^{-(v+1)/2}\) approaches \(e^{-t^2/2}\) and the gamma terms normalize similarly to \(\sqrt{2\pi}\).

Key concepts

Probability Density Function

The probability density function (PDF) is a fundamental concept in probability theory and statistics, especially when dealing with continuous random variables. Simply put, a PDF describes the likelihood of a random variable taking on a specific value.
For the t-distribution, which is a type of probability distribution, the PDF is defined with the number of degrees of freedom, denoted by \(v\). This distribution is useful when dealing with samples that are small or have an unknown variance.

The PDF for a t-distribution is given by:

• \( f(t) = \frac{\Gamma((v+1)/2)}{\sqrt{v\pi}\Gamma(v/2)} \left( 1 + \frac{t^2}{v} \right)^{-(v+1)/2} \)

Here, \(\Gamma\) represents the gamma function, which is a complex function that extends the factorial function to real and complex numbers. The constants and factors in this equation ensure that the area under the curve of the distribution equals 1, maintaining the property of total probability being 1.

Expectation and Variance

Expectation and variance are crucial metrics in statistics to understand the distribution of data. Expectation, \(\mathrm{E}(T)\), is the average or mean value expected from repeated trials of an experiment. For a t-distribution, this mean (or expectation) is zero, represented by \(\mathrm{E}(T) = 0\). This zero expectation is due to the symmetry of the t-distribution about the y-axis, provided that the degrees of freedom \(v > 1\).

Variance, represented as \(\operatorname{var}(T)\), measures how spread out the values of the random variable are around the mean. For the t-distribution:

• The variance is computed as \(\operatorname{var}(T) = \frac{v}{v-2}\) for \(v > 2\).

This variance is greater than 1, illustrating that t-distributions have heavier tails than the normal distribution, meaning they allow for a higher likelihood of values further from the mean.

Gamma Function

The gamma function (\(\Gamma(x)\)) plays a vital role in mathematics and statistics, particularly in defining distributions like the t-distribution. It generalizes the factorial function to non-integer values. While \(n!\) is defined as the product of all positive integers up to \(n\), \(\Gamma(x)\) is defined for real and complex numbers and extends this factorial concept.

The gamma function appears in the t-distribution formula, helping to define the shape and probability of the distribution. The function can be written as:

• \(\Gamma(n) = (n-1)!\) if \(n\) is a positive integer.
• For a non-integer, the gamma function is defined by an integral: \(\Gamma(x) = \int_0^\infty t^{x-1}e^{-t} \, dt\).

The gamma function ensures that the PDF is correctly normalized, so all probability sums up to 1.

Convergence to Normal Distribution

The concept of convergence to a normal distribution refers to how a series of distributions stabilizes to resemble the normal distribution as certain parameters approach infinity.
For the t-distribution, this convergence becomes evident as the degrees of freedom \(v\) increase. As \(v\) grows larger, the shape of the t-distribution becomes more like a standard normal distribution \(\phi(t)\), defined as:

• \(\phi(t) = \frac{1}{\sqrt{2\pi}}e^{-t^2/2}\)

This transformation is crucial as it allows for the use of normal distribution approximations in statistical analysis when sample sizes are large. As the degrees of freedom increase, the t-distribution's heavy tails (which indicate higher probabilities of extreme values) diminish, and it starts matching the bell curve characteristic of \(\phi(t)\), simplifying many practical statistical calculations.