### Exponential Distribution MLE Applet$X \sim exp(\lambda)$

 P(X > x) = P(X < x) = $\lambda=$ $x=$

This applet computes probabilities and percentiles for the exponential distribution: $$X \sim exp(\lambda)$$ It also can plot the likelihood, log-likelihood, asymptotic CI for $\lambda$, and determine the MLE and observed Fisher information.

#### Parameterization

• $f(x) = \lambda e^{-\lambda x}$ for $x>0$ (and 0 otherwise)
• $E(X) = 1/\lambda$
• $Var(X) = 1/\lambda^2$

#### Probability Density Function (pdf) mode

Directions:

• Enter the rate in the $\lambda$ box.
• Hitting "Tab" or "Enter" on your keyboard will plot the pdf.

To compute a right-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value, and press "Tab" or "Enter" on your keyboard. The probability $P(X \gt x)$ will appear in the pink box. Select $P(X \gt x)$ from the drop-down box for a left-tail probability (i.e. the cdf).

#### Maximum Likelihood (MLE) mode

Directions:

• Leave the $\lambda$ box empty.
• Enter the one or more data values in the $x$ box (separate multiple $x$ values by commas).
• Hitting "Tab" or "Enter" on your keyboard will plot the likelihood, log-likelihood, and 95% asymptotic CI for $\lambda$. The MLE and observed Fisher information are also displayed.

Note that the green line is drawn at $$max(log(likelihood))-\frac{3.84}{2}$$ The resulting asymptotic 95% CI is shown in green on the horizontal axis. The asymptotic CI is based on large sample theory; it is shown in this applet for all sample sizes, however.