Introduction to Probability · free preview
Universality of the Uniform
The Uniform as universal raw material: any distribution can be simulated from a Uniform, and any variable can be flattened back to a Uniform through its CDF.
1. The probability integral transform
The probability integral transform
2. From Uniform to anything: inverse transform sampling
From Uniform to anything: inverse transform sampling
3. Location-scale and standardization
Location-scale and standardization
4. The PIT direction
X has continuous strictly increasing CDF F. What is the distribution of F(X)?
5. Inverse transform sampling
To simulate a variable with target CDF F from a Uniform(0,1) draw U, what should you compute?
6. Exponential by inversion
F(x) = 1 - e^(-x) for x at least 0. F inverse of 0.5 equals ln 2, about what value? Enter like 0.69.
7. Standardizing a location-scale member
X has mean 10 and standard deviation 2 as a location-scale member of a standard family. The standardized value for x = 14 is (x minus mu) over sigma. Enter that number.
8. Uniform to Uniform
U is Uniform(0, 1). For X = 5 + 3U, what is P(X is at most 8)? Enter a decimal like 1.0.
9. In your own words
In your own words, state the most important definition, theorem or method of this lesson and explain what it says.
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