Visualizers
Distribution Explorer
Drag the parameters of the normal, lognormal, Poisson and binomial distributions and watch their shapes and stats respond.
Distribution Explorer
In finance: short-horizon log-returns are roughly normal, the bell curve behind mean–variance and Black–Scholes.
These are the handful of distributions that describe most of finance. The bell-shaped normal captures returns, the lognormal captures prices, since prices compound and can never fall below zero, while the Poisson counts events like trades per minute and the binomial counts successes over repeated yes/no trials. The parameters simply stretch and shift the shape: drag a slider and watch the mean, variance, and curve respond live.
Learn how it works
Five worked examples. Read a couple before you dive in, try to answer first, then reveal the solution.
Normal: same center, wider bell
Pick the Normal distribution. Compare μ = 0, σ = 1 against μ = 0, σ = 2.
Show solution
Both bells sit centered on 0, but the σ = 2 bell is shorter and wider, the same total area spread over a bigger range.
- μ (the mean) slides the whole bell left or right; it sets the center.
- σ (the standard deviation) sets the width, how far outcomes typically stray from the center.
In finance the Normal is the workhorse for daily returns and measurement errors: symmetric, with most values near the middle and rare extremes in the thin tails.
Lognormal: prices can't go negative
Switch to the Lognormal distribution and note its lopsided shape and where it starts.
Show solution
It's skewed to the right, a hump on the left, a long tail stretching to the right, and it never goes below 0.
That's exactly how prices behave:
- A stock can't fall below 0, but it can multiply many times over.
- Percentage moves compound, which naturally produces this right-skewed shape.
This is why models like Black-Scholes assume prices are lognormal while their percentage returns are Normal.
Poisson: counting rare events
Choose Poisson with λ = 3 and read off its mean and variance.
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You get a bumpy bar chart over the whole numbers 0, 1, 2, 3, …, it counts how many times something happens in a fixed window.
- Think trades per minute, orders hitting the book, or defaults in a year.
- A quirk: mean = variance = λ. Here both equal 3, so the average count is 3 and its spread is set by the same number.
Poisson is the go-to model for the arrival of rare, independent events.
Binomial: successes out of many tries
Pick Binomial with n = 20 trials and success chance p = 0.3. Check the mean.
Show solution
It counts how many successes you get out of 20 independent yes/no trials, like 20 coin-flip-style bets that each win 30% of the time.
- The peak sits at the mean = n × p = 20 × 0.3 = 6 successes.
- Values far from 6 (like 0 or 15) are possible but rare.
In finance it fits things like how many positions hit their target out of a batch, or wins out of a fixed number of trades.
Exponential: waiting time between events
Select Exponential and look at its shape, tall on the left, tail to the right.
Show solution
It models the waiting time between events, the gap until the next trade, the next price tick, the next default.
- Short waits are most common (the tall left side); long waits get rarer (the fading tail).
- Its mean = 1 ÷ λ, so a higher event rate λ means shorter typical waits.
- It's memoryless: having already waited a while tells you nothing about how much longer you'll wait, the clock effectively resets.
Exponential waits and Poisson counts are two views of the same event stream: one counts events, the other times the gaps.
What you'll learn
The shapes of the distributions that show up most in finance, how their parameters control them, and when each one applies.