Practice Questions 3

Mathematics for Economics and Finance Practice Questions III Instructor: Norman Schürho¤ Teaching Assistants: Zhihua (Cissy) Chen, Natalia Guseva Session: Fall 2006 Date: December 4, 2006 Problem 1 Suppose a monopolist’s marginal revenue is 6 2x where x denotes sales. Find the total revenue and the demand function faced by the monopolist. Problem 2 Three machines A, B, C produce resp. 50%; 30%; 20% of the total number of items of a factory. The percentages of defective items produced by the machines A, B, C are resp. 3%; 4%; and 5%: 1. If an item is selected at random, …nd the probability that the item is defective. 2. Suppose an item is selected at random and is found to be defective. Find the prob. that the item was produced by machine A Problem 3 Pick a card from a deck of 52. Let A be the event that “the card is an ace” and let B be the event that “the card is a spade”. Are events A and B independent? Problem 4 The pdf of normal distribution with mean and variance given by: (x )2 1 f (x; ; 2 ) = p e 2 2 ; 1<x<1 2 1. If X 2. Show

N( ; 1 R

1

p1 2

2

e

), show the random variable Z = z2 2

dz = 1;

1

X

2

is

is standard normal;

Problem 5 Standard quantitative models of the stock market assume stock return follows a log-normal distribution. If log X N ( ; 2 ); 0 < X < 1; 1 < < 1; 2 > 0: 1. Find the pdf for X; 2. Compute E(X); V ar(X): Problem 6 Let f (x) = 1=3 for 1 < x < 2 and zero elsewhere be the pdf for a random variable X: Find the pdf and cdf for the random variable Y = X 2 : 8 > > <

0 1=2 Problem 7 Suppose that the random variable X has cdf FX (x) = (x + 1) =2 > > : 1 Is X a discrete or continuous variable? Calculate the mean and variance of X: Problem 8 Consider the pdf de…ned by fX (x) = x23 ; x > 1 and zero elsewhere. Show (a) it’s correctly de…ned, (b) its expected value, (c) its variance. Problem 9 Let X and Y be independent random variables with means X ; Y and variances 2X ; 2Y : Find an expression for the correlation of XY and Y in terms of these means and variances. Problem 10 A median of a distribution is a value m such that P (X m) m 1 R R and P (X m) 12 : If X is continuous, m satis…es f (x)dx = f (x)dx = Show that minE jX a

aj = E jX

Problem 11 Show that if (X; Y ) the following are true:

1

mj :

Bivariate normal (

m

X;

Y

;

2 X;

2 Y

1 2 1 2

; ), then

(1) The marginal distribution of X is N ( X ; 2X );and the marginal distribution of Y is N ( Y ; 2Y ); Y (2) The conditional distribution of Y given X = x is N ( Y + ( X )(x 2 2 )): X ); Y (1 Problem 12 Take the regression model y = X n 1

n KK 1

+ " and assume that n 1

it ful…lls the main assumptions of the linear regression model. Furthermore, assume that y=X N (X ; 2 In ): (a). Write the log likelihood function. (b). Find the MLE estimators for and 2 : (c). Compute the Cramer-Rao lower bound and the Fisher Information matrix (d). Find the asymptotic distribution of the estimators.

2

if if if if

x<0 x=0 : 0<x 1 x>1

Problem 13 Let X1 ; :::; Xn represent a random sample from a normal Pndistribution with mean and variance 1. (a). Show that ^MpLE = Xn = n1 i=1 Xi is the MLE estimator and show that the variance of n(Xn p) is equal to p I( ) 1 = 1: (b). Calculate the probability of the event ^M LE 1:96 I( ) 1 = n < p p < ^M LE + 1:96 I( ) 1 = n:

Problem 14 Let X1 ; :::; Xn represent a random sample from N ( ; ML estimator b and prove its consistency.

2

): Derive

Problem 15 Y is a random variable that denotes the number of dots obtained when a fair six sided die is rolled. Let: X=

Y; if Y is even 0; otherwise

(i) Find joint distribution of (X; Y ) (ii) Find the best predictor of Y jX (iii) Find Best Linear Predictor of Y conditional on X (iv) Calculate Mean Squared Prediction Errors for cases (ii) and (iii).

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