Introduction to Probability · free preview
The Multivariate Normal Distribution
Vector-valued Normality parameterized by a mean vector and covariance matrix, correlation geometry, and conditional Normality with shrinkage.
1. Vectors of jointly Normal variables
Vectors of jointly Normal variables
2. Reading the parameters
Reading the parameters
3. Conditional Normality and shrinkage
Conditional Normality and shrinkage
4. Closure under linear maps
X is a bivariate Normal vector and W = 3 X_1 minus 2 X_2 plus 1. What is the distribution of W?
5. Zero correlation earns independence, here
A bivariate Normal has correlation 0. What follows?
6. Propagating a mean
X is Normal(2, 1) and Y is Normal(5, 4), independent. For W = X + 2Y, what is the mean of W? Enter an integer.
7. Propagating a variance
Continuing, Var(X) = 1 and Var(Y) = 4 independent. What is Var(W) for W = X + 2Y? Enter an integer.
8. Conditional variance shrinkage
Y has standard deviation 10 and correlation 0.6 with the observed X. The conditional standard deviation of Y given X is 10 times the square root of (1 minus rho squared). Enter that value rounded to one decimal, like 8.0.
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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