| | a1 | a2 |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ? |
1. What is ?(?=?2,?=?2)P(A=a2,B=b2)?
2. Observing events from this probability distribution,
what is the probability of seeing (a1, b1) then (a2,
b2)?
3. Calculate the marginal probability distribution,
?(?)P(A).
4. Calculate the marginal probability distribution,
?(?)P(B).
Consider the following collections of subsets of R:
B1 ={(a,∞):a∈R},
B2 ={(−∞,a):a∈R},
B3 ={[a,∞):a∈R},
B4 = {[a, b] : a, b ∈ R},
B5 = {[a, b] : a, b ∈ Q},
B6 ={[a,b]:a∈R,b∈Q}.
(i) Show that each of these is a basis for a topology on R.
(ii) What can you say about the corresponding topologies
T1,...,T6, eg, are any of the topologies the same, are any
comparable, are any equal to familiar topologies on R, etc?
a. The Log likelihood function is ?(?) = (a1 + a2) log(?) − ?(b1
+ b2) write this as a function of θ, by substituting in
θ = log(λ).
b. Write down the likelihood equation for θ, using the
log-likelihood in part a, and hence determine θ^ the MLE for θ.
c. Show that θˆlog = (λ^). Show this algebraically, what
property of MLEs is this?
d. Differentiate the LHS of the likelihood equation, obtain the
expected information ?(?) = ?{??(?,...
How do you Interpret the meaning of the different coefficients
(b0, b1, b2, b3,b4,…bn) in a multiple regression? (slightly
different from the interpretation in simple regression)
1. Three pairs of genes with two alleles each (A1 and A2, B1 and
B2, and C1 and C2) influence lifespan in a human population. The
alleles of these genes have an additive relationship and add the
number of years indicated to the lifespan of the individual.
allele
years
A1
15
A2
4
B1
16
B2
8
C1
13
C2
9
a. If lifespan were entirely genetically determined, what is the
minimum possible lifespan and the associated genotype?
b. If...
In Python
iOverlap (a1, a2, b1, b2)
Write the function iOverlap that tests whether 2 closed
intervals overlap. It takes 4 numbers (ints or floats) a1, a2, b1,
b2 that describe the two closed intervals [a1,a2] and [b1,b2] of
the real number line, and returns True if these two closed
intervals overlap (even if at only one point) and False otherwise.
If a1>a2, then the interval [a1,a2] is empty. If b1>b2, then
the interval [b1,b2] is empty. Both intervals are...
Let f: X→Y be a map with A1, A2⊂X and
B1,B2⊂Y
(A) Prove
f(A1∪A2)=f(A1)∪f(A2).
(B) Prove
f(A1∩A2)⊂f(A1)∩f(A2).
Give an example in which equality fails.
(C) Prove
f−1(B1∪B2)=f−1(B1)∪f−1(B2),
where f−1(B)={x∈X: f(x)∈B}.
(D) Prove
f−1(B1∩B2)=f−1(B1)∩f−1(B2).
(E) Prove
f−1(Y∖B1)=X∖f−1(B1).
(Abstract Algebra)