Question

In: Statistics and Probability

X1, X2, ..., Xnare mutually independent, following standard normal distribution N(0, 1). Define X as the...

  1. X1, X2, ..., Xnare mutually independent, following standard normal distribution N(0, 1). Define X as the sample mean...
    1. What is the distribution, mean and variance of X bar?
    2. What is the distribution, mean and variance of Xj-Xbar?
    3. Calculate the covariance of Xj and Xbar.

Solutions

Expert Solution

a.

By Central limit theorem, the distribution of X bar is Normal distribution with mean as population mean and variance = 1/n.

Thus, mean = 0

variance = 1/n

Thus, X bar ~ N(0, 1/n)

b.

We know that the linear combination of normal random variable is a normal random variable. Thus,   Xj-Xbar follows Normal distribution.

E[Xj-Xbar] = E[Xj] -E[Xbar] = 0 - 0 = 0

Var[Xj-Xbar] = Var[Xj] + Var[Xbar] - 2 Cov(Xj , Xbar) = 1 + 1/n - 2 Cov(Xj , Xbar)

Thus,

mean of Xj-Xbar = 0

Variance of Xj-Xbar = 1 + 1/n - 2 Cov(Xj , Xbar) =  1 + 1/n - 2/n (From part c, Cov(Xj , Xbar) = 1/n)

= 1 - 1/n

c.

Cov(Xj , Xbar) = Cov(Xj , (X1 + X2 + ... + Xj + ... + Xn )/n )

= (1/n) Cov(Xj , X1 + X2 + ... + Xj + ... + Xn )

= (1/n) [ Cov(Xj , X1) + Cov(Xj , X2) + .... + Cov(Xj , Xj) + .... Cov(Xj , X​​​n​) ]

= (1/n) [ 0 + 0 + .... + Var(Xj​​​​​​​) + .... 0 ] X1, X2, ..., Xnare mutually independent

= (1/n) * 1

= (1/n)


Related Solutions

Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼...
Let X1 and X2 be independent standard normal variables X1 ∼ N(0, 1) and X2 ∼ N(0, 1). 1) Let Y1 = X12 + X12 and Y2 = X12− X22 . Find the joint p.d.f. of Y1 and Y2, and the marginal p.d.f. of Y1. Are Y1 and Y2 independent? 2) Let W = √X1X2/(X12 +X22) . Find the p.d.f. of W.
Sample 1 has n1 independent random variables (X1, X2, ...Xn1) following normal distribution, N(µ1, σ21); and...
Sample 1 has n1 independent random variables (X1, X2, ...Xn1) following normal distribution, N(µ1, σ21); and Sample 2 has n2 independent random variables (Y1, Y2, ...Yn2 ) following normal distribution, N(µ2, σ22). Suppose sample 1 mean is X bar, sample 2 mean is Y bar , and we know n1 ,n2 , σ21 and σ22. Construct a Z statistic (i.e. Z~N(0, 1)) to test H0 : µ1 = µ2.
1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0...
1. Let X1, X2 be i.i.d with this distribution: f(x) = 3e cx, x ≥ 0 a. Find the value of c b. Recognize this as a famous distribution that we’ve learned in class. Using your knowledge of this distribution, find the t such that P(X1 > t) = 0.98. c. Let M = max(X1, X2). Find P(M < 10)
Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] =...
Suppose that random variable X 0 = (X1, X2) is such that E[X 0 ] = (µ1, µ2) and var[X] = σ11 σ12 σ12 σ22 . (a matrix) (i) Let Y = a + bX1 + cX2. Obtain an expression for the mean and variance of Y . (ii) Let Y = a + BX where a' = (a1, a2) B = b11 b12 0 b22 (a matrix). Obtain an expression for the mean and variance of Y . (ii)...
Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0.  elsewhere (a) Find the...
Let X1 and X2 have the joint pdf f(x1,x2) = 2 0<x1<x2<1; 0.  elsewhere (a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1 = x1. (b) Find the conditional expectation and variance of X1|X2 = x2 and X2|X1 = x1. (c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and P(0 < X1 < 1/2). (d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that var(Y ) ≤ var(X2).
Let X be a standard normal random variable so that X N(0; 1). For this problem...
Let X be a standard normal random variable so that X N(0; 1). For this problem you may want to refer to the table provided on Canvas. Recall, that (x) denotes the standard normal CDF. (a) Find (1:45). (b) Find x, such that (x) = 0:4. (c) Based on the fact that (1:645) = 0:95 nd an interval in which X will fall with 95% probability. (d) Find another interval (dierent from the one in (c)) into which X will...
Find the probability P(0<X1<1/3 , 0<X2<1/3) where X1, X2 have the joint pdf                    f(x1, x2)...
Find the probability P(0<X1<1/3 , 0<X2<1/3) where X1, X2 have the joint pdf                    f(x1, x2) = 4x1(1-x2) ,     0<x1<1 0<x2<1                                       0,                  otherwise (ii) For the same joint pdf, calculate E(X1X2) and E(X1 + X2) (iii) Calculate Var(X1X2)
Consider the independent observations x1, x2, . . . , xn from the gamma distribution with...
Consider the independent observations x1, x2, . . . , xn from the gamma distribution with pdf f(x) = (1/ Γ(α)β^α)x^(α−1)e ^(−x/β), x > 0 and 0 otherwise. a. Write out the likelihood function b. Write out a set of equations that give the maximum likelihood estimators of α and β. c. Assuming α is known, find the likelihood estimator Bˆ of β. d. Find the expected value and variance of Bˆ
If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36...
If the joint probability distribution of X1 and X2 is given by: f(X1, X2) = (X1*X2)/36 for X1 = 1, 2, 3 and X2 = 1, 2, 3, find the joint probability distribution of X1*X2 and the joint probability distribution of X1/X2.
The standard normal distribution is a continuous distribution with a mean of 0 and a standard...
The standard normal distribution is a continuous distribution with a mean of 0 and a standard deviation of 1. The following is true: About 68% of all outcomes lie within 1 St.Dev. About 95% of all outcomes lie within 2 St.Devs. About 99.7% of all outcomes lie within 3 St. Devs. The probability/percentage/percentile for a normal distribution is the area under the curve. You will ALWAYS BE CALCULATING OVER AN INTERVAL. Values outside of 2 St.Devs. are considered “unusual values.”...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT