Statistics shows Expected Value, Variance, Std Deviation and Mode for the current parameters; Show Formulas prints the rule under each card. Learn more about the statistics panel
PDF, the first tab above the chart, draws the density curve f(x); the red line marks E[X]. Learn more about the PDF curve
In the Probability Calculator, tap P(X < x), P(X ≤ x), P(X > x) or P(X ≥ x) and type x; strict and non-strict tabs give the same number. Learn more about cumulative probabilities
The calculator opens on P(a ≤ X ≤ b): type a and b for the area between them; the other three range tabs give the same area. Learn more about range probabilities
Use the λ (lambda) slider to set the event rate - how frequently events occur per unit time. Values range from 0.1 (rare events, long waits) to 5 (frequent events, short waits).
As λ increases, the PDF curve becomes steeper and more concentrated near zero, reflecting shorter average waiting times. Lower λ creates a more gradual decline, indicating longer typical waits between events.
The mean waiting time equals 1/λ. At λ = 2 events per hour, average waiting time is 0.5 hours. At λ = 0.5, expect 2-hour average waits. The relationship is perfectly reciprocal.
Reading the Statistics Panel
The Statistics panel sits in the left column, under Parameters. Its four cards show Expected Value, Variance and Std Deviation to four decimals, and Mode to four decimals. Every card recomputes on each slider move, so the panel always describes the curve in the middle column; the red E[X] line on the chart sits at the Expected Value number.
Show Formulas prints the rule under each card: E[X]=1/λ, Var(X)=1/λ2, σ=1/λ and Mode=0. The button then reads Hide Formulas. The numbers are the same either way; the formulas only show where they come from.
At the opening values, λ=1, the cards read 1.0000, 1.0000, 1.0000 and 0.0000. Expected Value and Std Deviation are equal, both 1/λ, and Mode is 0 at every λ. At λ=4 the cards read 0.2500, 0.0625, 0.2500 and 0.0000.
The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Rate Parameter (λ), Scale Parameter (1/λ) and Expected Value (E[X]). For where the formulas come from, see Distribution Statistics.
Reading the PDF Curve
DemoPDF, CDF and Table tabs
Step 0 of 4
The PDF shows the characteristic exponentially decreasing curve, starting at maximum height λ at x = 0 and declining toward zero as x increases. The curve never touches zero but approaches it asymptotically.
The steepness of the decline is controlled by λ. Higher rates create steeper drops, while lower rates produce more gradual decreases. The curve always has its mode at x = 0 - the shortest possible waiting time.
About 63% of the probability mass lies within one mean (1/λ) from zero. The practical display range extends to about 4/λ, capturing over 98% of all probability.
Understanding the CDF Display
The CDF shows P(X ≤ x) = 1 - e^(-λx), rising from 0 and approaching 1 asymptotically. Unlike the normal's S-curve, the exponential CDF rises quickly initially then gradually levels off.
The curve's initial steepness indicates high probability of short waiting times. At x = 1/λ (the mean), the CDF equals approximately 0.632 - about 63.2% of events occur within one average waiting time.
The CDF never quite reaches 1, reflecting the theoretical possibility of arbitrarily long waiting times, though probabilities become negligible for large x.
Using the Table Tab
Table is the third tab above the chart, after PDF and CDF. It replaces the chart with three columns, x, f(x) and F(x), one row per point the chart plots, every value to six decimals. The rows follow the sliders: move one and the whole table recomputes.
Use it where the chart is too small to read. At the opening values, λ=1, the table runs from x=0 to x=6.9078 in steps of 0.0230. The first row has f(0)=1.000000, equal to λ; the last row has F(x)=0.999000. Set λ=2 and the rows end at 3.4539, half as far.
The F(x) column is the CDF as numbers, and f(x) is the height of the PDF curve at that x. The density is not a probability; for one, use an interval. For a value the table does not list, or a range of values, use the Probability Calculator below the chart.
The Explorer at Its Opening Parameter
At λ=1 the density starts at its maximum, f(0)=λ=1, and decays from there. There is no interior peak: the most probable region is always immediately after zero.
The red line marks E[X]=1/λ=1, and the plotted window runs from 0 to about 6.91.
Exponential, lambda = 1: PDF above, CDF below
Density is largest at the very edge of the support, f(0) = 1, with no interior peak. The window ends at -ln(0.001) = 6.908, so the CDF below stops at exactly 0.999 by construction.
That window edge is chosen, not arbitrary. The tool uses xmax=−λln(0.001)=6.908, the point where the surviving tail is exactly one thousandth — so the CDF tab ends at 0.999 by construction rather than by accident.
The mean sits at 1 but the median is λln2=0.693, noticeably to its left. That gap is the signature of right skew, and it means "average waiting time" and "typical waiting time" are different questions with different answers.
Mean and standard deviation are both 1/λ, so the coefficient of variation is exactly 1 for every λ. Changing the rate rescales the whole picture without altering its shape — which is the memoryless property seen from the outside.
Computing Point Probabilities
For continuous distributions, individual point probabilities are always zero. What the curve shows instead is the density f(x) = λe^(-λx) — the height of the PDF, not a probability.
The PDF height indicates relative likelihood - higher values mean that region is more probable. The PDF at x = 0 equals λ, the maximum density, declining exponentially from there.
To find actual probabilities you need intervals. The Probability Calculator gives P(X ≤ x) (and P(X < x), P(X > x), P(X ≥ x)); an interval follows by subtraction, P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a), the area under the PDF curve between a and b.
Calculating Cumulative Probabilities
DemoCumulative probabilities
Step 0 of 4
Use P(X ≤ x) to find the probability of waiting x time units or less. The formula 1 - e^(-λx) provides exact values without numerical integration.
P(X > x) gives the survival probability - the chance of waiting more than x units. This equals e^(-λx), showing exponential decay. The memoryless property appears here: if you've waited time s, the probability of waiting additional time t is still e^(-λt).
P(X ≥ x) is identical to P(X > x) for continuous distributions since P(X = x) = 0. This differs from discrete distributions where the distinction matters.
Range Probability Calculations
DemoRange probabilities
Step 0 of 4
There is no separate range calculator: P(a ≤ X ≤ b) = F(b) - F(a) = e^(-λa) - e^(-λb), the difference of two P(X ≤ x) readings, giving the probability that waiting time falls between a and b.
The four boundary forms all give the same number for a continuous distribution, since individual points have zero probability: • [a, b], (a, b), [a, b), (a, b] all equal e^(-λa) - e^(-λb)
Example: For λ = 1, find P(0.5 ≤ X ≤ 2). This gives e^(-0.5) - e^(-2) ≈ 0.606 - 0.135 = 0.471, about 47% probability the wait is between 0.5 and 2 time units.
Key Properties and Real-World Applications
The right column holds two lists. Key Properties has 6 rows, each ending in ▼: Support: X ∈ [0, ∞), Memoryless Property, Constant Hazard Rate, Connection to Poisson Process, Decreasing Probability Density, Simple CDF and Quantiles. Row titles here are those at the opening values. Tap a row to open its explanation; tap another and the first closes, so one explanation is open at a time; tap the open row to close it.
Constant Hazard Rate and Connection to Poisson Process quote the current λ, so they change with the slider. Open a row, move a slider, and the numbers in the open explanation follow.
Real-World Applications, below, lists 6 situations the exponential distribution models, starting with Time until the next customer arrives at a store and Lifetime of electronic components (time until failure). This list is fixed text and does not change with the sliders.
The exponential distribution models waiting times between events in a Poisson process. If events occur randomly at constant rate λ, the time until the next event follows an exponential distribution.
The distribution requires constant event rate and independent occurrences. Each waiting period is independent of previous history - the memoryless property makes it unique among continuous distributions.
Applications include customer service times, equipment lifetimes, time between arrivals, radioactive decay intervals, and earthquake occurrences. For theoretical foundations, see exponential distribution theory page.
The Memoryless Property
The exponential is the only continuous distribution with the memoryless property: P(X > s+t | X > s) = P(X > t). If you've waited time s without an event, the probability of waiting additional time t is the same as starting fresh.
This means past waiting provides no information about future waiting. The distribution "forgets" how long you've already waited - each moment is statistically identical to any other.
The memoryless property arises from the constant hazard rate λ. Unlike aging processes where failure becomes more likely over time, exponential models truly random, time-independent events.
Distribution Statistics
The mean and standard deviation both equal 1/λ. With λ = 3 events per hour, average waiting time is 1/3 hour (20 minutes) with the same standard deviation.
The variance equals 1/λ², so variance equals mean squared. This high variance relative to the mean reflects the distribution's long right tail - occasional very long waits are possible.
The median equals (ln 2)/λ ≈ 0.693/λ, always less than the mean 1/λ. This reflects right skewness - the mode is at 0, median is lower than mean, creating positive skew.
Related Distributions and Tools
The Poisson distribution counts events in fixed time intervals, while exponential measures time between events. If events follow Poisson(λ) per unit time, inter-event times follow Exponential(λ).
The geometric distribution is the discrete analog, counting trials until first success rather than continuous waiting time. Both share the memoryless property.