Last modified: November 26, 2018

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Student's t-Distribution (Continuous)

The Student's t-distribution, or simply t-distribution, is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. The t-distribution is denoted as $X\sim t\left(\nu \right)$, where $\nu$ is the number of degrees of freedom.

Probability Density Function (PDF)

The PDF of a t-distribution is given by:

$$f\left(x\right)=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{(1+\frac{x^2}{\nu })}^{-\frac{\nu +1}{2}}$$

where $\Gamma (\cdot )$ is the gamma function.

Cumulative Distribution Function (CDF)

The CDF of a t-distribution can be expressed in terms of the regularized incomplete beta function, $I_x(a,b)$:

$$F\left(x\right)=1-\frac{1}{2}{I}_{\frac{x^2}{x^2+\nu }}(\frac{1}{2},\frac{\nu }{2})$$

Expected Value and Variance

The expected value (mean) of a t-distribution is equal to 0 for $\nu >1$:

$$E\left[X\right]=0$$

The variance of a t-distribution is given by:

$$\text{Var}\left(X\right)=\{\begin{array}{cc}\frac{\nu }{\nu -2},& \text{if}\nu >2\\ \infty ,& \text{if}1<\nu \le 2\\ \text{undefined},& \text{if}\nu \le 1\end{array}$$

Moment Generating Functions and Moments

The moment generating function (MGF) of a t-distribution does not exist due to the heavy tails of the distribution. However, we can compute the moments directly from the PDF.

• First Moment (Mean):

$$E\left[X\right]=0,\text{for}\nu >1$$

• Second Moment (Variance + Mean^2):

$$E\left[X^2\right]=\nu ,\text{for}\nu >2$$

Example: Analyzing the Effect of Diet on Blood Pressure

A researcher investigates whether two diets lead to different average systolic blood pressure. The following data is collected from a random sample:

• Diet A: Sample size (n1) = 20, Mean (mean1) = 120, Standard Deviation (s1) = 8
• Diet B: Sample size (n2) = 25, Mean (mean2) = 130, Standard Deviation (s2) = 10

To compare the means, a two-sample t-test is employed.

I. Calculation of the T-statistic:

The t-statistic is calculated using the formula for the two-sample t-test:

$$t=\frac{(\text{mean1}-\text{mean2})}{\sqrt{\left({\text{s1}}^2/\text{n1}\right)+\left({\text{s2}}^2/\text{n2}\right)}}$$

For the given data:

$$t=\frac{(120-130)}{\sqrt{\left(8^2/20\right)+\left({10}^2/25\right)}}\approx -3.78$$

This value indicates the number of standard deviations the sample mean difference is from 0.

II. Estimation of Degrees of Freedom:

The degrees of freedom can be estimated using the Welch-Satterthwaite equation:

$$df\approx \frac{\left(\right({\text{s1}}^2/\text{n1})+({\text{s2}}^2/\text{n2}){)}^2}{\left({\text{s1}}^4/\right({\text{n1}}^2(\text{n1}-1))+{\text{s2}}^4/({\text{n2}}^2(\text{n2}-1)\left)\right)}\approx 40.4$$

These degrees of freedom are used in determining the critical value of t from the t-distribution table for the hypothesis test.

Applications

Student's t-distributions are widely used in hypothesis testing (such as the one-sample t-test, two-sample t-test, and paired t-test), confidence interval estimation for the population mean, and in regression analysis. They are particularly useful when the sample size is small and the population standard deviation is unknown.