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4.9.8 · D5Probability Theory & Statistics

Question bank — Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta

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Before we start, one word we lean on constantly:

  • PDF = probability density function : height of the curve, NOT a probability.
  • CDF = cumulative distribution function : running area from the left.

The four little galleries below anchor the shapes the questions keep referring to — glance at them first so the words "hump", "U-shape", "J-shape", and "spike" mean something concrete.

Figure — Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta

True or false — justify

A "flat" PDF means every value is equally likely, so the probability of any single point is nonzero
False — for a continuous variable the probability of any single exact point is zero; "equally likely" refers to equal density, and probability only appears once you integrate over an interval.
For , doubling the width doubles the variance
False — variance is , so doubling the width multiplies variance by , not .
A PDF can be greater than at some point
True — only the area must equal ; e.g. has height everywhere. Density is not probability, so it faces no upper bound.
The exponential distribution's memorylessness means the event "resets" and becomes more likely the longer you wait
False — memoryless means the future wait has the same distribution regardless of elapsed time; it neither gets more nor less likely, it simply forgets.
If with integer , then is a sum of independent waits
True — Gamma is the wait for the -th event, which is the sum of successive first-event waits, each .
Because Normal is symmetric, always equals
True — symmetry about splits the total area of into two equal halves at the mean, for every and .
is just the uniform distribution on
True — plug : the density becomes , a flat line on .
A larger rate in means longer average waiting times
False — mean is , so a higher rate means events arrive faster and the average wait is shorter.
Beta's mean can exceed if is large
False — since , the fraction is always strictly between and , matching the support .
Two Normals with the same variance but different means have differently-shaped curves
False — changing only slides the curve horizontally; the shape (width, height) is set by alone, which is unchanged.

Spot the error

" because the range is and I halve it like a radius."
The correct answer is ; the comes from the algebra in , not from halving the range.
"For Normal I plug into the denominator, so the density is ."
Wrong — the ==standard deviation == (not ) sits in the normalizer and in the standardization .
" for is , I just read the table at 185."
You must standardize first: , then read ; tables only serve the standard .
"Exponential mean equals its parameter , since is the only knob."
The mean is the reciprocal ; is a rate (events per unit time), the mean is a time.
"Gamma has mean because both parameters push it up."
Mean is ; more required events () lengthen the wait, but a higher rate () shortens it, so divides.
"Beta leans right (toward 1) because 5 is bigger."
It leans left (toward 0): the larger piles "failure mass" near 0, giving mean .
"Since , the CDF's slope can be negative where the density dips."
The CDF is non-decreasing everywhere, so its slope ; density is never negative — a "dip" is a smaller positive slope, not a downhill CDF.

Why questions

Why does the exponential density come from the Poisson "zero events" probability?
"No event by time " is the survival (Poisson at ); differentiating gives the density. See Poisson Process.
Why is the magic constant in the Normal density?
It is the value of the Gaussian integral , so dividing by it forces the total area to .
Why does the Central Limit Theorem make the Normal appear everywhere?
Sums/averages of many independent small effects converge to a bell shape regardless of the pieces' own distributions — see Central Limit Theorem.
Why must every PDF integrate to exactly , not just be non-negative?
The total probability of all outcomes must be ; the area under is that total probability, so it is pinned to .
Why does the Beta function appear as the Beta density's denominator?
is exactly the raw area under the un-normalized shape, so dividing by it makes the density integrate to . See Beta Function.
Why does the Gamma function let us handle non-integer "number of events"?
smoothly interpolates the factorial () to all , so the Gamma distribution is defined even when isn't a whole count. See Gamma Function.
Why is exponential a special case of Gamma but not vice versa?
Setting in Gamma's density collapses to and gives ; the general Gamma with has no such collapse, so it's strictly more general.

Edge cases

Figure — Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta
Figure — Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta
What happens to as (interval shrinks to a point)?
The height while the width , so no ordinary PDF survives; the limit is a Dirac delta — an idealized "infinitely tall, zero-width spike" of unit area that is not a genuine function, marking the point where the continuous-density picture breaks down and the variable becomes deterministic.
What does do at ?
The density is at its maximum and decays from there; short waits are the most probable, and .
What shape does take when both ?
A U-shape with mass piling up at both endpoints and and thinning in the middle — the opposite of a central hump (blue curve in the figure above).
What shape does take when exactly one of is ?
A J-shape (or reversed-J): mass piles at the single endpoint whose exponent is negative — near if , near if (red curve above). Only that one end blows up.
What does look like near when ?
The density becomes singular (unbounded) at the origin, shooting to as , yet the area stays finite and equals ; for it is finite (), and for it starts at and rises to a hump.
What is the limiting behaviour of a Normal's tails as ?
The density but is never exactly zero; every real value has positive density, so a Normal technically has infinite support.
As Gamma's shape (with fixed), what does the distribution look like?
It becomes increasingly bell-shaped and symmetric, approaching a Normal — a direct echo of the Central Limit Theorem, since it's a sum of many exponentials. See Central Limit Theorem.
What does equal for any continuous distribution and any exact value ?
It equals ====, because a single point has zero width and ; only intervals carry positive probability.
What happens to the Beta mean as ?
The mean stays fixed at , but the variance , so the distribution ==concentrates tightly around ==.
Recall One-line summary of every trap

Density probability; rate mean; in the formula; standardize before reading tables; variance of uniform uses ; single points have probability zero; count the parameters below to know how many endpoints blow up.

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