Suppose f is a degree n polynomial where f(a1) = b1, f(a2) = b2, · ·...

Suppose f is a degree n polynomial where f(a1) = b1, f(a2) = b2, · · · f(an+1) = bn+1 where a1, a2, · · · an+1 are n + 1 distinct values. We will define a new polynomial g(x) = b1(x − a2)(x − a3)· · ·(x − an+1) / (a1 − a2)(a1 − a3)· · ·(a1 − an+1) + b2(x − a1)(x − a3)(x − a4)· · ·(x − an+1) / (a2 − a1)(a2 − a3)(a2 − a4)· · ·(a2 − an+1) + · · · + bn+1(x − a1)(x − a2)· · ·(x − an) / (an+1 − a1)(an+1 − a2)· · ·(an+1 − an).

Find g(a1), g(a2), · · · g(an+1).

Expert Solution

Colby Messinger answered 2 years ago

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1...

Prove: If a1 = b1 mod n and a2 = b2 mod n then (1) a1 + a2 = b1 + b2 mod n, (2) a1 − a2 = b1 − b2 mod n, and (3) a1a2 = b1b2 mod n.

What is Corr[A1, A2] where A1 equals B1 minus B2, and A2 equals B2 minus B3and...

What is Corr[A1, A2] where A1 equals B1 minus B2, and A2 equals B2 minus B3and we are given that Bi ∼ Ber(p) for all i in {1,2}?

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...

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)

| | a1 | a2 | |----|------|------| | b1 | 0.37 | 0.16 | | b2...

| | 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).

a. The Log likelihood function is ?(?) = (a1 + a2) log(?) − ?(b1 + b2) write...

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 ?(?) = ?{??(?,...

1. Three pairs of genes with two alleles each (A1 and A2, B1 and B2, and...

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...

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...

Gene A has two alleles, A1 and A2. Gene B has two alleles, B1 and B2....

Gene A has two alleles, A1 and A2. Gene B has two alleles, B1 and B2. In a population, the following haplotypes are observed. A1B1 = 0.2, A2B2 = 0.45, A1B2 = 0.15, and A2B1 = 0.2. Calculate the extent of linkage disequilibrium, D.

Translate the phrases to a system of linear equations: y= a1x+b1 and y=a2x+b2 with a variables,...

Translate the phrases to a system of linear equations: y= a1*x+b1 and y=a2*x+b2 with a variables, e.g. x and y. What do the y-intercepts represent in your example? What does the solution (or intersection) represent in your example? Solve the system of equation for x and y.

Show that the curve C(t) = <a1, a2, a3>t2 + <b1, b2, b3>t + <c1, c2,...

Show that the curve C(t) = <a1, a2, a3>t2 + <b1, b2, b3>t + <c1, c2, c3> lies in a plane and find the equation for such a plane.

Question