Compute
Prove the result is normal distributed. Obtain the mean and variance.
Compute Prove the result is normal distributed. Obtain the mean and variance. M(6353)d
prove that -slly distributed with mean- O and variance - 1, regardless ofthe particular values ofs and
If X1 and X2 are independent and identically distributed normal random variables with mean m and variance s2, find the probability distribution function for U=X1-3X2/2.
4. (10pt) Prove Ỳ, the vector of fitted values, is normally distributed with mean XB and variance oʻX(X'X)-'X'.
Assume X1, . . . , Xn iid normal with mean
and variance
^2 , show that
a. X¯ and X^2 are independent.
b. Proof that X¯ is normally distributed with mean
and variance
^2/n.
c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ?
1) degrees of freedom.
d. Show that X¯ S/is
t distributed with (n ? 1) degrees of freedom
by confidence intervals, normal distributed data, known
variance
Equation 1: If is the sample mean of a random sample of size n from a normal population with known variance o2, a 100 (1- a)% CI on u is given by HIZa/2 n SHST+/2 Vn is the upper 100g percentage point of the standard normal distribution. a/2 where 17. If the sample size n is doubled, by how much is the length of the CI on u in Equation 1 reduced?...
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Suppose that a rv Y has mgf m(t)- (a) 1-bt) Differentiate this mgf twice and thereby obtain the mean and variance of Y. [5 marksj] (b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t), ie·r(t) = login(1). Find r'() and r"(t), and express r'(0) and r"(0) in terms of EW and VarW. [5 marks] Use the result in (b) to find the mean (d) Find the mean and variance of the...
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y is the sum of independent random variables, compute both the mean and variance of Y. (b) Find the moment generating function of Y and use it to compute the mean and variance of Y.
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y...
Let X be a normal random variable with mean 0 and variance 0.5 and Y be exponentially distributed with mean 1. Suppose X and Y are independent. Find P(Y>X2 ).