PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Figure 9: Cauchy Distribution
(pointier than normal distribution and tails go to zero much slower than normal distribution)
If Z 1 ∼ N(0, 1) and Z 2 ∼ N(0, 1) independently, then X = Z 1
Z 2
∼ Cauchy distribution.
1.3 Transformations of a single RV
If X is a RV and Y = h(X), then Y is also a RV. If we know distribution of X, for a
given function h(x), we should be able to find distribution of Y = h(X).
Theorem. 1.3.1 (Discrete case)
Suppose X is a discrete RV with prob. function p X (x) and let Y = h(X), where h is
any function then:
p Y (y) =
∑
p X (x)
Proof.
x:h(x)=y
(sum over all values x for which h(x) = y)
p Y (y) = P (Y = y) = P {h(X) = y}
= ∑
x:h(x)=y
= ∑
x:h(x)=y
P (X = x)
p X (x)
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