Let Y be a random variable with p.d.f. ce-4y for y ≥ 0. (a) Determine c....
Let Y be a random variable with p.d.f. ce-4y for y ≥ 0.
(a) Determine c.
(b) What is the mean, variance, and squared coefficient of variation of Y where the squared
coefficient of variation of Y is defined to Var(Y )/(E[Y ])2?
(c) Compute Pr{Y < 5}.
(d) ComputePr{Y >5|Y >1}.
(e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that Pr{Y > q} = 0.7?
Homework Answers
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Let Y be a random variable with p.d.f. ce-4y for y ≥
0.
(a) Determine c....
6. Let Y be a random variable with p.d.f. ce-4y for y20 (a) Determine c. (b)...
6. Let Y be a random variable with p.d.f. ce-4y for y20 (a) Determine c. (b) What is the mean, variance, and squared coefficient of variation of Y where the squared coefficient of variation of Y is defined to Var(Y)/(E[Y)2? (c) Compute PríY < 5) (d) Compute PrY >5 |Y>1) (e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that Pr(Y > q} 0.7?
6. Let Y be a random variable with p.d.f. ce4y for y 2 0 (a) Determine...
6. Let Y be a random variable with p.d.f. ce4y for y 2 0 (a) Determine c. (b) What is the mean, variance, and squared coefficient of variation of Y where the squared coefficient of variation of Y is defined to Var(Y)/(E[Y])2? (c) Compute PríY <5) (d) Compute PríY >5 Y >1) (e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that PriY >
show steps thank you . Additional Problem 6. Let X be a continuous random variable with...
show steps thank you
. Additional Problem 6. Let X be a continuous random variable with pdf f(x) = (z + 1), -1 x 2. (a) Compute E(X), the mean of X. (b) Compute Var(X), the variance of X (c) Find an expression for Fx(), the edf of X. (d) Calculate P(X > 0). (e) Compute the mean of Y, where Y (f) Find mp, the pth quantile of X X-1 X+1
Suppose the c.d.f. of X is F(t) 3 for 0<t< (a) What is F(5)? (b) What...
Suppose the c.d.f. of X is F(t) 3 for 0<t< (a) What is F(5)? (b) What is F(-5)? (c) Compute the p.d.f of X. (d) Compute the mean of X (e) Compute the variance of X. (f) Compute the standard deviation of X (g) Compute the squared coefficient of variation of X.
Additional Problem 3. If X is a continuous random variable having cdf F, then its median...
Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) = 0.5. Find the median for random variables with the following density functions (a) f(r)-e*, x > 0 (c) f(x) 6r(1-x), 1. Additional Problem 6. Let X be a continuous random variable with pdf (a) Compute E(X), the mean of X (b) Compute Var(X), the variance of X. (c) Find an expression for Fx(r),...
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] =...
Let X ~ N(0, 1) and let Y be a random variable such that E[Y|X=x] = ax +b and Var[Y|X =x] = 1 a) compute E[Y] b) compute Var[Y] c) Find E[XY]
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y<...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
For a random walk with random starting value, let Y, Yoterter-1e for t > 0, where...
For a random walk with random starting value, let Y, Yoterter-1e for t > 0, where Yo has a distribution with mean μ0 and variance σό . Suppose fur- ther that Yo, et.., e are independent. (a) Show that E(Y) Ho for all t. (b) Show that Var(,) = tơ24 (c) Show that Cody, Y.) = min(t, s) + 05, , lienee , that cov ( var (it) and (d) Show that Corr(,) = 1 for 0st s s.
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0...
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
5. Let X be a Poisson random variable with parameter λ = 6, and let Y...
5. Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute EY
6. Let Y be a random variable with p.d.f. ce-4y for y20 (a) Determine c. (b) What is the mean, variance, and squared coefficient of variation of Y where the squared coefficient of variation of Y is defined to Var(Y)/(E[Y)2? (c) Compute PríY < 5) (d) Compute PrY >5 |Y>1) (e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that Pr(Y > q} 0.7?
6. Let Y be a random variable with p.d.f. ce4y for y 2 0 (a) Determine c. (b) What is the mean, variance, and squared coefficient of variation of Y where the squared coefficient of variation of Y is defined to Var(Y)/(E[Y])2? (c) Compute PríY <5) (d) Compute PríY >5 Y >1) (e) What is the 0.7 quantile (or 70th percentile) where the 0.7 quantile is the point q such that PriY >
show steps thank you
. Additional Problem 6. Let X be a continuous random variable with pdf f(x) = (z + 1), -1 x 2. (a) Compute E(X), the mean of X. (b) Compute Var(X), the variance of X (c) Find an expression for Fx(), the edf of X. (d) Calculate P(X > 0). (e) Compute the mean of Y, where Y (f) Find mp, the pth quantile of X X-1 X+1
Suppose the c.d.f. of X is F(t) 3 for 0<t< (a) What is F(5)? (b) What is F(-5)? (c) Compute the p.d.f of X. (d) Compute the mean of X (e) Compute the variance of X. (f) Compute the standard deviation of X (g) Compute the squared coefficient of variation of X.
Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) = 0.5. Find the median for random variables with the following density functions (a) f(r)-e*, x > 0 (c) f(x) 6r(1-x), 1. Additional Problem 6. Let X be a continuous random variable with pdf (a) Compute E(X), the mean of X (b) Compute Var(X), the variance of X. (c) Find an expression for Fx(r),...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
For a random walk with random starting value, let Y, Yoterter-1e for t > 0, where Yo has a distribution with mean μ0 and variance σό . Suppose fur- ther that Yo, et.., e are independent. (a) Show that E(Y) Ho for all t. (b) Show that Var(,) = tơ24 (c) Show that Cody, Y.) = min(t, s) + 05, , lienee , that cov ( var (it) and (d) Show that Corr(,) = 1 for 0st s s.
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
5. Let X be a Poisson random variable with parameter λ = 6, and let Y = min(X, 12). (a) What is the p.m.f. of X? (b) What is the mean of X? (c) What is the variance of X? (d) What is the p.m.f. of Y? (e) Compute EY
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