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Exercise 3 (Paired test, known variance of the difference). Let X, Y...
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X]...
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,Y) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1,..., Xn-Ynare i.i.d. with normal distribution, show that (exactly)...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some...
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some distribution with mean u and finite variance. Define the sample variance S2 = (n-1)-1 _, (Xi - X)2. We will show that S2 is an unbiased estimator of the population variance Var(X1). (i) Show that ) = 0. (ii) Show that [ŠX – 1908–) -0. ElCX –po*=E-* (Šx--) == "Varex). x:== X-X+08 – ) Lx - X +2Zx - XXX - 1) + X...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
3. Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume...
3. Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is Nux, ox) and Y is Nuy, o). Given that nx = 25, i = 34.68, $ = 4.88; ny = 29, y = 32.55 and 5 = 5.81, test the null hypothesis Ho: Mix = My against H :MX > Hy with a =0.01. Hint: you need to justify whether it is reasonable to assume same variance for the two samples...
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X,...
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X, Y be two Bernoulli random variables and let P a r = = = P(X = 1) (the probability that X = 1) P(Y = 1) (the probability that Y = 1) P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1), ... ,(Xn, Yn) be a sample of n i.i.d. copies of (X, Y)....
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform o...
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the...
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
Happiness and Being in a Relationship Let X, Y be two Bernoulli random variables and let...
Happiness and Being in a Relationship Let X, Y be two Bernoulli random variables and let p = P(X = 1) (the probability that X = 1) q = P(Y = 1) (the probability that Y = 1) r = P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1),...,(Xn, Yn) be a sample of n i.i.d. copies of (X,Y). Define and do not assumer = pq. = 3 X.Y.. We...
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,Y) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1,..., Xn-Ynare i.i.d. with normal distribution, show that (exactly)...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
Exercise 5 (Sample variance is unbiased). Let X1, ... , Xn be i.i.d. samples from some distribution with mean u and finite variance. Define the sample variance S2 = (n-1)-1 _, (Xi - X)2. We will show that S2 is an unbiased estimator of the population variance Var(X1). (i) Show that ) = 0. (ii) Show that [ŠX – 1908–) -0. ElCX –po*=E-* (Šx--) == "Varex). x:== X-X+08 – ) Lx - X +2Zx - XXX - 1) + X...
Exercise 5.23. Let (Xn)nz1 be a sequence of i.i.d. Bernoulli(p) RVs. Let Sn -Xi+Xn (i) Let Zn-(Sn-np)/ V np (1-p). Show that as n oo, Zn converges to the standard normal RV Z~ N(0,1) in distribution. (ii) Conclude that if Yn~Binomial(n, p), then (iii) From i, deduce that have the following approximation x-np which becomes more accurate as n → oo.
3. Let X and Y denote the tarsus lengths of male and female grackles, respectively. Assume that X is Nux, ox) and Y is Nuy, o). Given that nx = 25, i = 34.68, $ = 4.88; ny = 29, y = 32.55 and 5 = 5.81, test the null hypothesis Ho: Mix = My against H :MX > Hy with a =0.01. Hint: you need to justify whether it is reasonable to assume same variance for the two samples...
Bookmark this page Setup: All problems on this page will follow the definitions here: Let X, Y be two Bernoulli random variables and let P a r = = = P(X = 1) (the probability that X = 1) P(Y = 1) (the probability that Y = 1) P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1), ... ,(Xn, Yn) be a sample of n i.i.d. copies of (X, Y)....
Exercise 4.8: Suppose that X1, X2,..., Xn is a random sample of observations on a r.v. X, which takes values only in the range (0, 1). Under the null hypothesis Ho, the distribution of X is uniform on (0, 1), whereas under an alternative hypothesis, њ, the distribution is the truncated exponential with p.d.f. 0e8 where 6 is unknown. Show that there is a UMP test of Ho vs Hi and find, roximately, the critical region for such a test...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
Happiness and Being in a Relationship Let X, Y be two Bernoulli random variables and let p = P(X = 1) (the probability that X = 1) q = P(Y = 1) (the probability that Y = 1) r = P(X = 1, Y = 1) (the probability that both X = 1 and Y = 1). Let (X1,Y1),...,(Xn, Yn) be a sample of n i.i.d. copies of (X,Y). Define and do not assumer = pq. = 3 X.Y.. We...
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