Continuous distributions: uniform, normal and logistic

What is a continuous uniform distribution?

A continuous uniform distribution makes every value between a and b equally likely. The density is a flat line at 1/(b − a), and the cumulative curve is a straight line from 0 to 1.

$$\mu = \dfrac{a+b}{2}, \quad \sigma^2 = \dfrac{(b-a)^2}{12}$$

$$P(x_1 \le X \le x_2) = \dfrac{x_2 - x_1}{b - a}$$

How does a uniform draw turn into any distribution?

Draw u between 0 and 1, then find the outcome whose cumulative probability equals u. That is the inverse of the cumulative distribution function, and the method is called inverse transform sampling . It is how a simulation produces draws from any distribution.

A Monte Carlo simulation of a share price works the same way. It draws u, converts it to a standard normal draw with the inverse of the normal cumulative distribution function (CDF), and feeds it into a price model (the next lesson, and Learning Module 9).

What is the normal distribution?

The normal distribution is a symmetric bell fully described by its mean μ and variance σ², written N(μ, σ²). Mean, median and mode are equal. Its skewness is 0 and its kurtosis is 3 (excess kurtosis 0). The standard normal distribution has μ = 0 and σ = 1.

$$P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.68, \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.95$$

The normal appears everywhere because the sum of many small independent effects tends to a normal. Confidence intervals and tests in Learning Module 7 build on it. There is no table lookup in this lesson.

What is the logistic distribution?

The logistic distribution is a symmetric bell with an S-shaped cumulative curve. It has a location parameter μ, which is its mean, and a scale parameter s. Its variance is s²π²/3, its skewness is 0, and its kurtosis is 4.2 (excess kurtosis 1.2), so its tails are heavier than the normal’s.

$$E[X] = \mu, \quad \sigma^2 = \dfrac{s^2 \pi^2}{3}$$

The logistic distribution is the one behind logistic regression, which models yes/no outcomes such as default or no default, and it turns up in machine learning and in growth processes that level off. Poisson regression, by contrast, models counts.

How do the four moments compare?

The four moments of a distribution are its mean (center), variance (spread), skewness (lopsidedness) and kurtosis (tail weight). Learning Module 5 measured them from data. Here they come from the distribution’s formulas.

Distribution	Mean	Variance	Skewness	Kurtosis (excess)
Continuous uniform [a, b]	(a + b)/2	(b − a)²/12	0	1.8 (−1.2)
Normal N(μ, σ²)	μ	σ²	0	3 (0)
Logistic (μ, s)	μ	s²π²/3	0	4.2 (1.2)

All three are symmetric. What separates them is the tails: none for the uniform, moderate for the normal, heavy for the logistic.

Continuous distributionssmooth curves; probability is area

• Continuous uniform Memorize flat, no tails; kurtosis 1.8
• Normal Memorize symmetric bell; kurtosis 3
• Logistic Memorize symmetric bell, heavier tails; kurtosis 4.2
• Log-normal Use it next lesson: skewed right, bounded at zero

Try the Uniform tab first, then the inverse draw. On the Normal tab change μ and σ to see where the 68% and 95% bands fall. On the third tab both curves share the same mean and standard deviation, so any difference you see is in the tails; zoom in on them.

How do you do this in a spreadsheet?

Know the idea

Task	Excel and Google Sheets
Uniform draw between a and b	RAND()*(b − a) + a
Normal density / cumulative probability	NORM.DIST(x, mean, sd, FALSE) / NORM.DIST(x, mean, sd, TRUE)
Normal draw (inverse CDF of a uniform draw)	NORM.INV(RAND(), mean, sd); Google Sheets also accepts NORMINV
Uniform cumulative probability	(x − a)/(b − a), or BETA.DIST(x, 1, 1, TRUE, a, b)

Go deeper The normal density and distribution function in symbols

The density is f(x) = [1 ÷ (σ√(2π))] × e^(−½((x − μ) ÷ σ)²), and the cumulative function is F(x) = Φ((x − μ) ÷ σ), where Φ is the standard normal cumulative function. You do not need to memorize either. In this module, normal probabilities come from the 68% and 95% ranges; a spreadsheet’s NORM.DIST does the rest.

Go deeper Tails in numbers

With the same standard deviation, the chance of a move beyond three standard deviations in either direction is about 0.27% for the normal and about 0.86% for the logistic. That is the practical meaning of excess kurtosis of 1.2: more extreme moves than the normal predicts. These figures are for intuition, not for the exam.

Go deeper Why logistic regression uses this curve

A logistic regression turns any score into a probability between 0 and 1 by passing it through the S-shaped cumulative curve. That is why it suits yes/no questions such as “will this borrower default?”. Counts, such as the number of defaults in a quarter, call for Poisson regression instead.