How to Find an Exponential Percentile on a TI-84

The TI-84 can calculate an exponential percentile directly from the inverse cumulative distribution formula. This is useful because the standard distribution menus do not generally include a dedicated inverse-exponential command. First check whether the problem gives the exponential distribution’s rate or its mean, then enter the appropriate expression on the calculator’s home screen.

Use the inverse CDF

For an exponential random variable with rate $\lambda$ , the cumulative distribution function is

F(x)=P(X\le x)=1-e^{-\lambda x},\qquad x\ge 0.

The $p$ th percentile is the value $x_p$ for which $F(x_p)=p$ . Solve that equation for $x_p$ :

\begin{aligned} p&=1-e^{-\lambda x_p}\\ e^{-\lambda x_p}&=1-p\\ -\lambda x_p&=\ln(1-p)\\ x_p&=-\frac{\ln(1-p)}{\lambda}. \end{aligned}

Here, $p$ is a decimal cumulative probability: use $0.90$ for the 90th percentile, for example. The result is in the same units as $X$ .

Check whether the parameter is a rate or mean

Textbooks and problems use more than one parameterization. If the distribution is written with rate $\lambda$ , its mean is $1/\lambda$ . If the problem gives the mean $\mu$ instead, then $\lambda=1/\mu$ , and the formula becomes

x_p=-\mu\ln(1-p).

For example, if the mean waiting time is 5 minutes, use $\mu=5$ minutes. Do not divide by 5: the rate is $1/5=0.2$ per minute. Both parameterizations give the same percentile when converted correctly.

TI-84 example: find the 90th percentile

Suppose waiting time is exponentially distributed with rate $\lambda=0.2$ per minute. For the 90th percentile, set $p=0.90$ :

x_{0.90}=-\frac{\ln(1-0.90)}{0.2} =-\frac{\ln(0.10)}{0.2} \approx 11.5129\text{ minutes}.

On the TI-84 home screen, enter:

(-) ln(1 - .90) ÷ .2

Then press ENTER. Use the calculator’s negation key (-) for the negative sign at the start of the expression. The result should be about 11.51292546; rounding to two decimal places gives 11.51 minutes.

If the problem instead says the mean is 5 minutes, enter:

(-) 5 × ln(1 - .90)

This also gives about 11.5129 minutes. These expressions use the ln key, not the common logarithm log key.

If the problem gives an upper-tail percentage

A percentile refers to the proportion of observations at or below a value. Thus, the 90th percentile leaves 10% above it. If a problem asks for the cutoff with 10% of waiting times above it, use $p=1-0.10=0.90$ in the formula. In general, if the stated upper-tail probability is $q$ , enter $p=1-q$ .

For a percentile stated as a percentage, convert it to a decimal first: the 75th percentile uses $p=0.75$ . The formula gives finite interior percentiles for $0<p<1$ . The 0th percentile is $0$ , while the 100th percentile is not finite because the exponential distribution has unbounded support.

Common mistakes

Using the percentage as a whole number: enter .90, not 90, for the 90th percentile.
Using $p$ instead of $1-p$ inside the logarithm: the inverse CDF uses $\ln(1-p)$ . For a high percentile, $1-p$ is a small positive number.
Confusing rate and mean: if the mean is 5, the rate is 0.2, not 5. Use either $-\ln(1-p)/\lambda$ with the rate or $-\mu\ln(1-p)$ with the mean.
Reversing the tail: “10% above” corresponds to the 90th percentile, not the 10th percentile.
Ignoring units: a rate measured per hour produces a percentile in hours. Convert the final result if the question asks for a different time unit.