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![Mean and Variance of a Uniform Distribution Using the definitions of expectation and variance leads to the following calculations. As you might expect, for a uniform distribution, the calculations are not difficult. Using the basic definition of expectation we may write: b2- a2 2(b - a) b+ a 2 Using the formula for the variance, we may write: V(X) E(X2)-[E(X)]2 = b+ a (0-a)2 12](http://img.homeworklib.com/questions/1948e730-7547-11ea-81a7-19ee03991de2.png?x-oss-process=image/resize,w_560)
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1. Using calculus, find the mean and variance of a uniform distribution with a minimum value...
2. Using calculus, find the mean and variance of an exponential distribution with a probability density...
2. Using calculus, find the mean and variance of an exponential distribution with a probability density function of f(x)-Aet, L? (Give a proof What is A in terms of
1) Suppose X is a Normal RV with mean = 12 and variance = 16. Find...
1) Suppose X is a Normal RV with mean = 12 and variance = 16. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 16 or X > 12). Hint: Remember to always identify outcomes of interest first! (d) The center of the probability density function of X.
(1 point) Find the maximum and minimum values of the function f(x, y) = 3x² –...
(1 point) Find the maximum and minimum values of the function f(x, y) = 3x² – 18xy + 3y2 + 6 on the disk x2 + y2 < 16. Maximum = Minimum =
7. (1 point) Let X be the mean of a random sample of size 36 from...
7. (1 point) Let X be the mean of a random sample of size 36 from the uniform distribution U(7,15) Find P(11.3 <X < 11.5)
2) Suppose X is a Normal RV with mean = 17 and variance = 4. Find...
2) Suppose X is a Normal RV with mean = 17 and variance = 4. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 11 or X > 17) (d) P(X < 11 and X > 17)
The random variable Z has a Normal distribution with mean 0 and variance 1. Show that...
The random variable Z has a Normal distribution with mean 0 and variance 1. Show that the expectation of Z given that a < Z < b is o(a) – °(6) 0(b) – (a)' where Ø denotes the cumulative distribution function for Z.
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance...
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points
(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) =
3. Using calculus, find the mean of a normal distribution with a probability density function of(Give...
3. Using calculus, find the mean of a normal distribution with a probability density function of(Give a proof.)
Central limit theorem 9. Suppose that a random variable X has a continuous uniform distribution fx(3)...
Central limit theorem 9. Suppose that a random variable X has a continuous uniform distribution fx(3) = (1/2,4 <r <6 o elsewhere (a) Find the distribution of the sample mean of a random sample of size n = 40. (b) Calculate the probability that the sample mean is larger than 5.5.
2. Using calculus, find the mean and variance of an exponential distribution with a probability density function of f(x)-Aet, L? (Give a proof What is A in terms of
1) Suppose X is a Normal RV with mean = 12 and variance = 16. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 16 or X > 12). Hint: Remember to always identify outcomes of interest first! (d) The center of the probability density function of X.
(1 point) Find the maximum and minimum values of the function f(x, y) = 3x² – 18xy + 3y2 + 6 on the disk x2 + y2 < 16. Maximum = Minimum =
7. (1 point) Let X be the mean of a random sample of size 36 from the uniform distribution U(7,15) Find P(11.3 <X < 11.5)
2) Suppose X is a Normal RV with mean = 17 and variance = 4. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 11 or X > 17) (d) P(X < 11 and X > 17)
The random variable Z has a Normal distribution with mean 0 and variance 1. Show that the expectation of Z given that a < Z < b is o(a) – °(6) 0(b) – (a)' where Ø denotes the cumulative distribution function for Z.
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points
(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) =
3. Using calculus, find the mean of a normal distribution with a probability density function of(Give a proof.)
Central limit theorem 9. Suppose that a random variable X has a continuous uniform distribution fx(3) = (1/2,4 <r <6 o elsewhere (a) Find the distribution of the sample mean of a random sample of size n = 40. (b) Calculate the probability that the sample mean is larger than 5.5.
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