GEOMETRY OF DATA
EXAMPLES FOR STUDY & PRACTICE
1. One-sided chebyshev: For Z ∈ L2 with EZ =0& varZ = 1 verify that, for any t ≥ 0:
P(Z>t) ≤ 1/(1 + t2).
There are at least three di↵erent ways to arrive at this result:
(a) Hint: Z>t (Z + s)2 ≥ (t + s)2 s > 0. Apply markov.
(b) Hint: t − Z ≤ (t − Z)I(Z ≤ t). Apply cauchy-schwarz.
(c) Hint: (1 + t
2)2I(Z>t) ≤ (tZ + 1)2.
2. wP1-equality
For any X ∈ R we say that iff P(X = Y ) = 1.
(a) Verify that is an equivalence relation on R.
(b) Verify
(c) Verify
(d) Verify
3. Verify that, in L2, P is an orthogonal projection (onto W = JmP)
if and only if
it has the following three properties:
and, in particular, on Rn, P is an orthogonal projection
4. For an orthogonal projection, P, with P +Q = I, verify that
Use this result, or otherwise, to verify that
from which we have the special case of pythagorus
5. For nested sub-spaces V < W in L2, if Q is the orthogonal projection onto V, while P is the orthogonal projection onto W, verify that
6. Suppose that X ~ bin(2, 1/3), Y ~ poisson(2/3) and X Y .
a) Given that , determine k.
b) Determine the ratio ||X−Y||/||X−EX||.
c) Determine the coefficient of correlation ρ(X−Y,X+Y ).
7. Suppose that X ~ N(1, 1), let Y = X3, and consider the simple
linear model
Y = α+βX + W w. EW =0= ρ(X,W).
a) Evaluate the constants α and β.
b) Determine the relative proximity of Y to its closest linear predictor
8. Let X ~ exp (1), Y = e−X, and consider the simple
linear model
Y = α+βX + W w. EW =0= ρ(X,W).
a) Evaluate the constants α, β and γ.
b) Determine the relative proximity of Y to its closest linear predictor