![3. Suppose a value is chosen at random in the interval [0,6]. In other words, r is an observed value of a random variable X U(0,6). The random variable X divides the interval [0,6] into two subintervals, the lengths of which are X and 6- X respectively. Denote by Y min(X, 6-X), the length of the shorter one of the two intervals. Find e probability PY> ) for any given y. Then find both the cdf and the pdf of Y Can you tell the name of the distribution of Y?](http://img.homeworklib.com/questions/5b64dad0-044f-11ec-8b6f-697f11947096.png?x-oss-process=image/resize,w_560)
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3. Suppose a value is chosen "at random" in the interval [0,6]. In other words, r...
Suppose that a point X is selected at random from the interval (0,1). After the value X = x has been selected, a point Y is then chosen at random from the interval (0,x^2). a) Indicate the region R on...
Suppose that a point X is selected at random from the interval (0,1). After the value X = x has been selected, a point Y is then chosen at random from the interval (0,x^2). a) Indicate the region R on the xy-plane of possible values of the random vector (X,Y). b) Find the marginal pdf f2(y) of Y.
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
X is a positive continuous random variable with density fX(x). Y = ln(X). Find the cumulative...
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
1. Suppose that X and Y are random variables that can only take values in the intervals 0 X 2 and...
1. Suppose that X and Y are random variables that can only take values in the intervals 0 X 2 and 0 Y 3 2. Suppose also that the joint cumulative distribution function (cdf) of X and Y, for 0 < 2 and 03 y 3 2, is as follows: Fy). 16 [5] (a) Determine the marginal cdf Fx(x) of X and the marginal cdf Fy () of Y [5] (b) Determine the joint probability density function (pdf) f(x, y)...
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outpu...
Please answer the question thoroughly.
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outputs -2 if the chosen number falls in the interval [0, 1/4). It outputs 1 if the chosen number falls in the interval [1/2,2/3). Otherwise, the random variable iply outputs the chosen number. Find the distribution function Fx associated to X, find its discrete and continuous parts, Fxd and Fxe, and draw their graphs.
Exercise 4.10. A number is...
Part 1 Suppose that 2 batteries are randomly chosen without replacement from a group of 12...
Part 1 Suppose that 2 batteries are randomly chosen without replacement from a group of 12 batteries: 3 new, 4 used (working), and 5 defective. Let the random variable X denote the number of new batteries chosen and the random variable Y denote the number of used batteries chosen. The joint distribution fxy is given in the following table: 0 12 17663/6 120/6612/66 1. Calculate P ( X 1 ,Y > 1) 2. Find the marginal probability mass function fx...
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0....
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0. Let Y = √X. a. Find the cdf for Y. b. Find the pdf for Y. c. Find E[Y]. If you want to skip a difficult integration by parts, make a substitution and look for a Gamma pdf. d. This Y is actually a commonly used continuous distribution. Can you name it and identify its parameters? e. Suppose that X is exponentially distributed with...
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then...
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X. Uniform (a,...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ unifo...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
Please answer both. . Suppose that Y is a random variable with distribution function below. 1-e-v/2,...
Please answer both.
. Suppose that Y is a random variable with distribution function below. 1-e-v/2, 0, y > 0; otherwise F(y) = (a) Find the probability density function (pdf) f(y) of Y. yso (b) E(Y) and Var(Y) 5. Suppose X is a random variable with E(X) 5 and Var(X)-2. What is E(X)?
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
X is a positive continuous random variable with density fX(x). Y
= ln(X).
Find the cumulative distribution function (cdf) Fy(y) of Y in terms of the cdf of X. Find the probability density function (pdf) fy(y) of Y in terms of the pdf of X. For the remaining problem (problem 3 (3),(4) and (5)), suppose X is a uniform random the interval (0,5). Compute the cdf and pdf of X. Compute the expectation and variance of X. What is Fy(y)?...
1. Suppose that X and Y are random variables that can only take values in the intervals 0 X 2 and 0 Y 3 2. Suppose also that the joint cumulative distribution function (cdf) of X and Y, for 0 < 2 and 03 y 3 2, is as follows: Fy). 16 [5] (a) Determine the marginal cdf Fx(x) of X and the marginal cdf Fy () of Y [5] (b) Determine the joint probability density function (pdf) f(x, y)...
Please answer the question thoroughly.
Exercise 4.10. A number is chosen uniformly from the interval [0, 1). The random variable X outputs -2 if the chosen number falls in the interval [0, 1/4). It outputs 1 if the chosen number falls in the interval [1/2,2/3). Otherwise, the random variable iply outputs the chosen number. Find the distribution function Fx associated to X, find its discrete and continuous parts, Fxd and Fxe, and draw their graphs.
Exercise 4.10. A number is...
Part 1 Suppose that 2 batteries are randomly chosen without replacement from a group of 12 batteries: 3 new, 4 used (working), and 5 defective. Let the random variable X denote the number of new batteries chosen and the random variable Y denote the number of used batteries chosen. The joint distribution fxy is given in the following table: 0 12 17663/6 120/6612/66 1. Calculate P ( X 1 ,Y > 1) 2. Find the marginal probability mass function fx...
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X. Uniform (a,...
2. Suppose that the CDF of X is given by Fur :53 e-3 for x <3 Fx)for 3 for r >3. 1 (a) Find the PDF of X and specify the support of X. (b) Given a standard uniform random variable U ~ uniform(0, 1), find a transformation g) so that X g(U) has the above CDF. (Hint: This entails the quantile function F-().)
2. Suppose that the CDF of X is given by Fur :53 e-3 for x 3....
Please answer both.
. Suppose that Y is a random variable with distribution function below. 1-e-v/2, 0, y > 0; otherwise F(y) = (a) Find the probability density function (pdf) f(y) of Y. yso (b) E(Y) and Var(Y) 5. Suppose X is a random variable with E(X) 5 and Var(X)-2. What is E(X)?
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