Binomial Approximation and Joint Distributions Chris Piech CS109, Stanford University Which is Random? Climate Sensitivity Four Prototypical Trajectories
Review The Normal Distribution
2 • X is a Normal Random Variable: X ~ N(µ, s ) § Probability Density Function (PDF): 1 2 2 f (x) = e-(x-µ) / 2s where - ¥ < x < ¥ s 2p µ
§ E[X ] = µ f (x) 2 § Var(X ) =s x § Also called “Gaussian” § Note: f(x) is symmetric about µ Simplicity is Humble
µ
probability σ2
value
* A Gaussian maximizes entropy for a given mean and variance Density vs Cumulative
CDF of a Normal 1 F(x)
PDF of a Normal f(x)
0 -5 5 f(x) = derivative of probability F(x) = P(X < x) Probability Density Function
2 (µ, ) the distance to N “exponential” the mean
(x µ)2 1 f(x)= e 2 2 p2⇡
probability sigma shows up density at x a constant twice Cumulative Density Function
2 CDF of Standard Normal: A (µ, ) function that has been solved N for numerically x µ F (x)= ✓ ◆ The cumulative density function (CDF) of any normal
Table of F(z) values in textbook, p. 201 and handout Four Prototypical Trajectories
Great questions! Four Prototypical Trajectories
68% rule only for Gaussians? 68% Rule?
What is the probability that a normal variable X N(µ, 2) ⇠ has a value within one standard deviation of its mean?
µµ µµ XX µµ µµ++ µµ P (µ