1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y =...
1. Consider the uniform distribution X defined over the interval [0, 2pi]. Now let Y = sin(X)
(a) Calculate the CDF FY(y) of Y.
(b) Calculate the PDF f(y) of Y. In particular, in what interval [a, b] is Y defined? (this mean f(y) = 0 for y < a and for y > b).
(c) Verify that f(y) is a PDF.
Homework Answers
ANSWER:
Given that,
a)
The cdf of 
is
b)
The pdf of
is 
. Hence
is defined in the interval 
.
c)
To verify that 
is a pdf,
Also 
. Hence a pdf.
Add Answer to:
1. Consider the uniform distribution X defined over the interval
[0, 2pi]. Now let Y =...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
Let X be an uniform distribution between 0 and 1, Y be an uniform distribution between...
Let X be an uniform distribution between 0 and 1, Y be an uniform distribution between -5 and 3, and they are independent. Calculate the pdf, expectation, and the variance of the followings 1. 4X
Suppose that Y=cos(X), where X is uniformly distributed over the interval [0, 2Pi]. Determine the pdf...
Suppose that Y=cos(X), where X is uniformly distributed over the interval [0, 2Pi]. Determine the pdf of the random variable Y.
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a)...
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a) The PDF fY (y) b) The mean E[Y ] c) Here X is uniform in (0, 1), but now you are asked to find a functiong(·) such that the PDF of Y = g(X) is ?2y 0≤y<1fY (y) = 0 otherwise
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution...
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution of Y given X = x be U(0, x3) i. Determine f (x, y). Make sure to describe the support of f. ii. Calculate fy (y) iii. Find E(Y).
15. (10 points) A. Draw a graph of the probability distribution function (PDF) for the uniform di...
15. (10 points) A. Draw a graph of the probability distribution function (PDF) for the uniform distribution that is defined to be non-zero and constant between 1 and 10. Label the x and y-axes for the graph. (3 points) B. On the same graph draw the cumulative distribution function (CDF) for the uniform distribution. Clearly identify each line (PDF or CDF) in the graph. (3 points) C. In words, express the mathematical relationship that exists between any CDF and the...
Verify that if Y has a beta distribution with a = B = 1, then Y...
Verify that if Y has a beta distribution with a = B = 1, then Y has a uniform distribution over (0, 1). That is, the uniform distribution over the interval (0, 1) is a special case of a beta distribution. If a = B = 1 then r(C ), -y , sys f(y) = , f(1) elsewhere. Note that (2) = and f(1) = . Thus, after simplification and evaluating the gamma functions the pdf of Y becomes fly)...
5.2 Square law detector - continued. Continue to consider Example 5.2, in which Y = g(x) = x2 (a) Let X have a uniform...
5.2 Square law detector - continued. Continue to consider Example 5.2, in which Y = g(x) = x2 (a) Let X have a uniform distribution over [-1, +1]. Find the distribution function and PDF of the square-law detector output Y. (b) Let X be a Gaussian variable with zero mean and variance 02; i.e., Ex(x) = ¢ () and fx(x) = 54(5, (5.88) where 0 (u) and (u) are the distribution function and PDF of the unit normal variable U...
2. Assume the random variable y has the continuous uniform distribution defined on the interval a...
2. Assume the random variable y has the continuous uniform distribution defined on the interval a to b, that is, f(y) = 1/6 - a), a sy<b. For this problem let a = 0 and b = 2. (a) Find P(Y < 1). (Hint: Use a picture.) (b) Find u and o2 for the distribution.
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
Let X be an uniform distribution between 0 and 1, Y be an uniform distribution between -5 and 3, and they are independent. Calculate the pdf, expectation, and the variance of the followings 1. 4X
(c) (20 pts.) Let X have a uniform distribution U(0, 2) and let the considiton; distribution of Y given X = x be U(0, x3) i. Determine f (x, y). Make sure to describe the support of f. ii. Calculate fy (y) iii. Find E(Y).
15. (10 points) A. Draw a graph of the probability distribution function (PDF) for the uniform distribution that is defined to be non-zero and constant between 1 and 10. Label the x and y-axes for the graph. (3 points) B. On the same graph draw the cumulative distribution function (CDF) for the uniform distribution. Clearly identify each line (PDF or CDF) in the graph. (3 points) C. In words, express the mathematical relationship that exists between any CDF and the...
Verify that if Y has a beta distribution with a = B = 1, then Y has a uniform distribution over (0, 1). That is, the uniform distribution over the interval (0, 1) is a special case of a beta distribution. If a = B = 1 then r(C ), -y , sys f(y) = , f(1) elsewhere. Note that (2) = and f(1) = . Thus, after simplification and evaluating the gamma functions the pdf of Y becomes fly)...
5.2 Square law detector - continued. Continue to consider Example 5.2, in which Y = g(x) = x2 (a) Let X have a uniform distribution over [-1, +1]. Find the distribution function and PDF of the square-law detector output Y. (b) Let X be a Gaussian variable with zero mean and variance 02; i.e., Ex(x) = ¢ () and fx(x) = 54(5, (5.88) where 0 (u) and (u) are the distribution function and PDF of the unit normal variable U...
2. Assume the random variable y has the continuous uniform distribution defined on the interval a to b, that is, f(y) = 1/6 - a), a sy<b. For this problem let a = 0 and b = 2. (a) Find P(Y < 1). (Hint: Use a picture.) (b) Find u and o2 for the distribution.
Most questions answered within 3 hours.
-
Choose an enzyme and tell its function and what substrate it
works on.
asked 1 minute ago -
a) Given these data for the reaction , write the rate-law
expression.
(Use k for the...
asked 1 minute ago -
How many moles of oxygen would be consumed during the aerobic
oxidation of one mole of...
asked 2 minutes ago -
Compare two countries how the culture
of those country enhances or impedes your understanding of the...
asked 2 minutes ago -
Identify the incorrect statement.
A. An increase in mol does not increase the average force of...
asked 4 minutes ago -
In broadcast authentication, a senderaims to send
oneauthentication code for multiple receivers to verify. Explain
why...
asked 4 minutes ago -
A student studying for a vocabulary test knows the meanings of
16 words from a list...
asked 35 minutes ago -
List the most common organic functional groups that function as
acids or bases. Write an equation...
asked 16 minutes ago -
1) A protein is 435 amino acids long, which of the genes below
COULD NOT code...
asked 19 minutes ago -
What do the phenomena of overshadowing, the CS preexposure
effect, and relative validity of cues have...
asked 47 minutes ago -
Assuming air has a density of 1.17 g/L and .973 atm what is the
average molar...
asked 40 minutes ago -
Developmental Biology! Please answer all the questions
7) Mislocalization of oscar to the side of the...
asked 34 minutes ago