Consider the probability distribution ?(?) = ??n, 0 ≤ ? ≤ 1 for a positive integer...
-
Consider the probability distribution ?(?) = ??n, 0 ≤ ? ≤ 1 for a positive integer ?.
-
Derive an expression for the constant ?, to normalize ?(?).
-
Compute the average 〈?〉 as a function of ?.
-
Compute the expectation value of the second moment.
-
Compute the variance as a function of ?.
-
Homework Answers
![(3) <^2) = E (a) = (*_*(ntvmdn ht) nt3 and order moment (4) variance = Ln²> - [<ny] ht3 nta, = (n+1)* {(n+2)2-(n+1)(n+3)? (n+](http://img.homeworklib.com/questions/bf06a1a0-b5f5-11eb-8e10-a36c4a409f29.png?x-oss-process=image/resize,w_560)
Add Answer to:
Consider the probability distribution ?(?) = ??n, 0 ≤
? ≤ 1 for a positive integer...
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer.
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n]...
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n] = _ for n=1, 2, ,. Random variables, 偶; j=1, 2, . ) do not depend on N and are independent with the common exponential distribution, with the mean equal to θ 2, or equivalently (2)-, e-0.5x, for x 〉 0. Consider a random sum, 1. Derive the marginal expectation of S 2. Derive the marginal variance of S. 3. Find the marginal second...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
Consider a particle confined to one dimension and positive z with the wave function 0 where...
Consider a particle confined to one dimension and positive z with the wave function 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: f) Calculate the expectation value of the momentum, (p) via the canonical expression -0o g) Calculate the expectation value of (p) via the canonical expression h) Use your results for(i) and (pay to calculate the variance in...
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use...
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
(a) The probability distribution for the height h of molecules of mass m in our atmosphere,...
(a) The probability distribution for the height h of molecules
of mass m in our atmosphere, assuming a constant temperature, is
given by
P(h) ∝ e-mgh/kBT
Normalize the probability distribution and derive an expression
for the average height of molecules in the atmosphere.
(b) Estimate the height of Oxygen molecules in the atmosphere,
and of Nitrogen molecules.
You may use the result
Consider a particle confined to one dimension and positive r with the wave function 0, z<0...
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find...
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
Consider the probability mass distribution PſY = 1] = c and PſY = k] = k...
Consider the probability mass distribution PſY = 1] = c and PſY = k] = k · P[Y = k – 1] for k = 2, ...5. for some constant c. a) Calculate c. b) Calculate the probability that Y is an odd number. b) Calculate expectation and variance of Y.
3. Let N = R and P be the probability distribution on (R, B(R)) with density...
3. Let N = R and P be the probability distribution on (R, B(R)) with density 1 XER. Put X(w):=w2, WEN. (a) Describe o(X) (the o-algebra on 2 generated by X). Justify your answer. (b) Derive the distribution function Fx of the random variable X. (c) Compute the mean EX and moment generating function y(t) := t> 0, for the random variable (RV) X. [3+3+3=9] EetX 300,
You are given that a random variable, N, has the geometric distribution, 3-1 PIN = n] = _ for n=1, 2, ,. Random variables, 偶; j=1, 2, . ) do not depend on N and are independent with the common exponential distribution, with the mean equal to θ 2, or equivalently (2)-, e-0.5x, for x 〉 0. Consider a random sum, 1. Derive the marginal expectation of S 2. Derive the marginal variance of S. 3. Find the marginal second...
Consider a particle confined to one dimension and positive with the wave function Nxear, x20 x<0 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: a) Find the normalization constant N. What are the units of your result and do they make sense? b) What is the most probable location to find the particle, or more precisely, at what z...
Consider a particle confined to one dimension and positive z with the wave function 0 where N is a real normalization constant and α is a real positive constant with units of (length)-1. For the following, express your answers in terms of α: f) Calculate the expectation value of the momentum, (p) via the canonical expression -0o g) Calculate the expectation value of (p) via the canonical expression h) Use your results for(i) and (pay to calculate the variance in...
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
(a) The probability distribution for the height h of molecules
of mass m in our atmosphere, assuming a constant temperature, is
given by
P(h) ∝ e-mgh/kBT
Normalize the probability distribution and derive an expression
for the average height of molecules in the atmosphere.
(b) Estimate the height of Oxygen molecules in the atmosphere,
and of Nitrogen molecules.
You may use the result
Consider a particle confined to one dimension and positive r with the wave function 0, z<0 where N is a real normalization constant and o is a real positive constant with units of (length)-1. For the following, express your answers in terms of a: a) Calculate the momentum space wave function. b) Verify that the momentum space wave function is normalized such that (2.4) c) Use the momentum space wave function to calculate the expectation value (p) via (2.5)
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
Consider the probability mass distribution PſY = 1] = c and PſY = k] = k · P[Y = k – 1] for k = 2, ...5. for some constant c. a) Calculate c. b) Calculate the probability that Y is an odd number. b) Calculate expectation and variance of Y.
3. Let N = R and P be the probability distribution on (R, B(R)) with density 1 XER. Put X(w):=w2, WEN. (a) Describe o(X) (the o-algebra on 2 generated by X). Justify your answer. (b) Derive the distribution function Fx of the random variable X. (c) Compute the mean EX and moment generating function y(t) := t> 0, for the random variable (RV) X. [3+3+3=9] EetX 300,
Most questions answered within 3 hours.
-
Which of these four researchers is most likely to have made Type
1 error
Kim who...
asked 5 minutes ago -
Create ER Model
Assets are resources that are used in CMS to display content.
Each asset...
asked 5 minutes ago -
In a performance test, each of two cars takes 8.9 s to
accelerate from rest to...
asked 6 minutes ago -
Sapphire Aerospace operates 52 weeks per year, and its cost of
goods sold last year was...
asked 9 minutes ago -
A. Suppose your manager indicates that for a normally
distributed data set you are analyzing, your...
asked 10 minutes ago -
For each the following molecules draw the best possible
structure based on formal charges for each...
asked 10 minutes ago -
A proton moves in the plane of this paper toward the top of the
page. A...
asked 20 minutes ago -
different divisions differing lines of business use
different costs of capital because their cost of equity...
asked 16 minutes ago -
Give an algorithm to determine whether or not the elements
of
an array of integers are...
asked 23 minutes ago -
. Gravimetric analysis. The gravimetric analysis that we will
perform in lab occurs according to the...
asked 42 minutes ago -
Learning curve problem:
A company was ordered to make 6 custom made parts. the labor
costs...
asked 39 minutes ago -
When you don’t recall operator precedences you can Guess
Experiment with the compiler Look in a...
asked 42 minutes ago