### Student's t-Distribution$X \sim t_{(\nu)}$

 $\nu=$ $x=$ P(X > x) = P(X < x) = 2P(X > |x|) = P(-|x| < X < |x|) =

This applet computes probabilities and percentiles for the t-distribution: $$X \sim t_{(\nu)}$$

#### Directions:

• Enter the degrees of freedom in the $\nu$ box.
• To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
• To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your keyboard. The percentile $x$ will appear in the blue box.

On the graph, the $x$ value appears in blue while the probability is shaded in pink.

#### Details

• Probability density function $$f(x)=\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\ \Gamma(\frac{\nu}{2})} \left( 1+\frac{x^2}{\nu} \right )^{-\frac{\nu+1}{2}}$$ where $-\infty < x < \infty$ and $\nu > 0$
• $\mu=E(X)=0$ for $\nu>1$
• $\sigma^2=Var(X)=\frac{\nu}{\nu-2}$ for $\nu>2$
• $\sigma=SD(X)=\sqrt{\frac{\nu}{\nu-2}}$ for $\nu>2$