The CLT limit. Sums of many small independent things land here. Z-scores, confidence intervals, and Black-Scholes all live in the Normal world.
Method · Normal
Intro
M. W. Toews, CC BY 2.5 · CC-BY-2.5 · Wikimedia Commons
Normal is the CLT limit of summed bounded-variance noise. The next question β 'what about the PRODUCT of many such noises?' β leads to the Lognormal.
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
IQ scores in a population are modelled as Normal with mean ΞΌ = 100 and standard deviation Ο = 15. (a) Compute P(85 < X < 115). (b) Compute P(X > 130). (c) Find the IQ score that exactly 95% of the population falls below.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
More examples
A handful of harder problems on the same theme. Click any prompt to reveal the solution sketch.
Quick warm-up on conditional expectation. Suppose $X$ is a standard normal random variable, $X \sim N(0, 1)$. What is $E[X \mid X > 0]$?
$E[X \mid X > 0] = \sqrt{2/\pi} \approx 0.798$. Compute $E[X \cdot \mathbb{1}\{X > 0\}] = \int_0^\infty x \cdot \tfrac{1}{\sqrt{2\pi}} e^{-x^2/2}\, dx = \tfrac{1}{\sqrt{2\pi}}$. Divide by $P(X > 0) = 1/2$ to get $\sqrt{2/\pi}$.
+ More examples Β· expand
Reflection
Where does this distribution sit in the story chain β what question does it answer that the previous distribution couldn't? Try to recall the key moment formulas (mean, variance) without looking them up.
