Introduction to Probability · free preview
Convolutions and Sums of Independent Variables
The convolution formula for densities of independent sums, worked examples from Uniforms to Exponentials and Normals, and the exponential family's self-convolution.
1. The convolution formula
The convolution formula
2. Convolving named distributions
Convolving named distributions
3. Checking and using convolutions
Checking and using convolutions
4. Convolution limits
X and Y are independent Uniform(0,1), and T = X + Y. For t = 1.5, the convolution integral runs over which x-range?
5. Recognizing a convolution shortcut
Which sum should be computed by a named-distribution shortcut rather than direct convolution?
6. Triangle peak
For two independent Uniform(0,1) sums, the density at t = 0.5 equals t, that is 0.5. What is the density at t = 1, the peak? Enter like 1.0.
7. Exponential sum mean
X and Y are independent Expo(2). The sum X + Y has mean E(X) + E(Y). Enter a decimal like 1.0.
8. Gamma from repeated convolution
Summing 5 independent Expo(1) variables gives a Gamma distribution with shape parameter 5. What is its rate parameter? Enter an integer.
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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