Last modified: November 04, 2024

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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 $X \sim t(\nu)$, where $\nu$ is the number of degrees of freedom.

Probability Density Function (PDF)

The PDF of a t-distribution is given by:

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

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, $I_x(a, b)$:

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

Expected Value and Variance

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

$$E[X] = 0$$

The variance of a t-distribution is given by:

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

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):

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

• Second Moment (Variance + Mean^2):

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

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:

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

For the given data:

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

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:

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

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.