Show that the probability for two random polynomials in Z[r] of degree at mostn and max-norm at most A to be coprime in Q[r] is at least 1- 1/(2A+1).

If аn аssessment is essentiаl оr рerсeived benefiсiаl in аny mаnner, аuthоrs аre likely…

Answered: Show that the probability for two…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

Prove that the maximal entropy of a discrete random variable is logan (n being the number of possible values of the random variable) and is attained for P₁ = P2 = = P₁ = 1/n.
Determine the Z-transform of the following DT signals
1- Let S(x) denote the quadratic spline polynomials. What is S(7/2) for the data set {(0, 0), (3, 1), (4, 2)}. 1/3 5/6 5/3 7/3 10/3 a) b) C) d) e) 2002099
Consider a Poisson process of intensity λ > 0 events per hour. Suppose that exactly one event occurs in the first hour. Let X be the time at which this event occurs, measured in hours (so that the possible values of X are the reals in the interval [0, 1]). (a) What is P (0 ≤ X ≤ a) for a ∈ [0, 1]? (Of course, we are conditioning on exactly one event occurring in the first hour, as described.) Hint: Consider the number of events that occur in [0, a] and in (a, 1], similar to question #5 on the previous homework. (b) What is the distribution of X, under the same conditions? (You might want to wait until Monday’s class after break to do this last part.)
Two points are selected randomly on a line segment of length 20, one on each side of the midpoint of the line. That is, the two points X and Y are independent random variables such that X is uniformly distributed over (0, 10) and Y is uniformly distributed over (10, 20). Find the probability that the distance between the two points is greater than 8. Hint: Think of the pair (X, Y) as a point on the x-y plane (as in the Mr. Johnson and the newspaper problem) and identify the region where they're more than 8 apart..
Let (S,F) be the corresponding measurable space. Consider the random variable X2 defined as by X₂ (w) = 2 if w EDS 1 if w ED\Ds 0 if w= miss Where D={(x,y) = R^2: x^2 +y^2< 1} and Ds={(x,y)=R^2:x^2+y^2≤ 1/4}. Describe the o-algebra generated by X, i.e., what is σ(X)?
Suppose that the joint density function of two random variables W₁ and W₂ is given by F (w₁, W₂) = {C(W₁ + w?), 0, 0
3. Let the random variable X have the pdf f(x) = 2(1 — x), 0 ≤ x ≤ 1, zero elsewhere. a) Find the cdf of X. Provide F(x) for all real numbers x (set up the appropriate cases). b) Find P(1/4 < X < 3/4). c) Find P(X= 3/4). d) Find P(X ≥ 3/4).
i) Let X be an integrable random variable on (2, F, P) and G be a sub o-field of F. State the definition of the conditional expectation of X given G.
Let X = (X1, X2, ..., Xn)T be a random vector with mean vector μ and covariance matrix Σ. Suppose that for every a = (a1, a2, ..., an)T ∈ Rn, the random variable aTX has a (one-dimensional) Gaussian distribution on R. a) For fixed a ∈ Rn, compute the moment generating function MaTX(u) of the random variable aTX, writing the answer using μ & Σ. b) Define MX(t1, t2, ... , tn), the moment generating function of the random vector X, and express it in terms of MtTX, the moment generating function of tTX where t = (t1, t2, ..., tn). c) Combining a) and b), compute the moment generating function MX(t) of X and hence prove that the random vector X has a Gaussian distribution.
Let X₁, X₂,... be a sequence of binomial random variables on a probability space (S,F), P) with probability mass function Show that P(X₂ = 1) = 1 - P(X₂ = 0) = Xn a.s. → 0. 1 n2, n ≥ 1.
Let XXX be uniformly distributed on [1,3][1,3][1,3]. What is the expectation of X2X^2X2, i.e., E[X2]E[X^2]E[X2]? Provide at least two decimal places.

SEE MORE QUESTIONS

Recommended textbooks for you

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability (10th Edition)

Probability

ISBN:

9780134753119

Author:

Sheldon Ross

Publisher:

PEARSON

A First Course in Probability

Probability

ISBN:

9780321794772

Author:

Sheldon Ross

Publisher:

PEARSON

SEE MORE TEXTBOOKS

Show that the probability for two random polynomials in Z[r] of degree at most n and max-norm at most A to be coprime in Q[r] is at least 1- 1/(2A+ 1).