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How distributions connect
A connection between two distributions is useful only when you know what operation makes it true. Adding independent variables, transforming one variable, changing a parameter, and averaging over an uncertain parameter are different operations.
Choose counts and waiting times or the normal family. The numbered map provides orientation; the list supplies the precise conditions and links. On small screens, the list carries the whole explanation without shrinking diagram labels.
Read each arrow together with its numbered condition below. A transformation constructs a new variable; a mixture averages conditional distributions; a special case is exact; a limit requires a parameter sequence.
1. Bernoulli → Binomial transformation
Sum n independent Bernoulli(p) variables with the same p. The count is Binomial(n,p); unequal probabilities generally give a different law.
2. Binomial → Poisson limit
As n grows and p shrinks with np tending to λ>0, Binomial(n,p) converges in distribution to Poisson(λ). Finite-n accuracy depends on the event being calculated.
3. Exponential → Gamma transformation
Sum k independent exponential waits with one common rate λ. For positive integer k the sum is Gamma(shape k, rate λ), or scale 1/λ. Noninteger gamma shapes are not literal counts of waits.
4. Gamma → Exponential special case
Gamma(shape 1, rate λ) is exactly Exponential(rate λ). In the scale convention, both use scale 1/λ.
Use a connection to solve a problem
Three independent waits at rate two events per minute add to a gamma wait with shape 3 and rate 2. Its mean is 3/2 minutes. Doubling the rate halves this mean; it does not change the number of events you are waiting for. The connection requires a common rate: arbitrary exponential waits with different rates do not add to this gamma model.
For a second example, a normal variable with mean zero and variance one becomes log-normal after exponentiation. The median is 1, but the mean is exp(1/2), about 1.649. A nonlinear transformation changes how an average behaves; see Jensen’s inequality.
Make a prediction
Does the arrow from Student’s t to normal mean you can substitute a normal tail probability at three degrees of freedom?
Explore the answer
No. That arrow describes a limit as degrees of freedom increase. At three degrees of freedom, the two-sided probability beyond two scale units is about 0.1393, compared with 0.0455 for standard normal. Use the actual distribution for a finite parameter value.
Continue with a concrete event
Use Poisson vs binomial or normal vs Student’s t to measure approximation error for the same event. If the mechanism is still unclear, the distribution chooser identifies assumptions to investigate first.