Concept explainers
Suppose that
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
EBK AN INTRODUCTION TO MATHEMATICAL STA
- The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.arrow_forwardConsider two assets (X and Y) with mX = 10%, mY = 10%, σX2=.16, σY2=.25, and Cov(X,Y) = -.125. What is the expected return and variance of the portfolio having 70% invested in X and 30% invested in Y? Compare the risk and return of this portfolio with the risks and returns associated with investing everything in either X or Y. a) What is p(XY)? b) What is the expected return of the portfolio (m.7X+.3Y)? c) What is the standard deviation of the portfolio (s.7X+.3Y)? d) How does the standard deviation of the portfolio (s.7X+.3Y) compare to the standard deviations of assets X and Y? Show all work and formulas used in EXCELarrow_forwardSuppose that a continuous random variable, X, is defined over the interval [0,1]. The Cdf of X is given by F(x)=x2 , what is Pr(X<0.2)? a). 0.25 b). 0.16 c).0.04 d). 0.1875arrow_forward
- The moment generating function of the random variable X is given by mX(s) = e2e^(t)−2 and the moment generating function of the random variable Y is mY (s) =(3/4et +1/4)10. If it is assumed that the random variables X and Y are independent, findthe following:(a) E(XY)(b) E[(X − Y )2](c) Var(2X − 3Y)arrow_forwardLet Xi be arandom sample from U(0,1)prove that Xn’ convarges in probability to 0.50arrow_forwardLet m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?arrow_forward
- A certain brand of upright freezer is available in threedifferent rated capacities: 16 ft3, 18 ft3, and 20 ft3. LetX = the rated capacity of a freezer of this brand sold ata certain store. Suppose that X has pmf x 16 18 20 p(x) 0.2 0.5 0.3 a. Compute E(X), E(X2), and V(X). b. If the price of a freezer having capacity X is70X - 650, what is the expected price paid by thenext customer to buy a freezer? c. What is the variance of the price paid by the nextcustomer? d. Suppose that although the rated capacity of a freezeris X, the actual capacity is h(X) = X - 0.008X2. Whatis the expected actual capacity of the freezer purchased by the next customer?arrow_forwardLet X be a random variable with probability mass function P ( X = 1 ) = 1/2 , P ( X = 2 ) = 1/3 , a n d P ( X = 5 ) = 1/6 . Then E[1/x]=?arrow_forwardLet Z be a N(0, 1) random variable, and let F(x) be thecumulative distribution function for Z. Show that on S (∞, 0], F(x) is an increasing convex function, and on S [0, ∞), F(x) is an increasing concave function.arrow_forward
- 1)Let x be a uniform random variable over the interval (0, 1). Knowing that y = x2 , calculate:a)Determine Fy(Y) = P(y<=Y),Y real and determine the pdf of y.b)Calculate E[x2] , using the pdf of x.c)Calculate E[y], using the pdf of y and compare with part (b).arrow_forwardLet X be a continuous random variable described below: p(X) = x/4, for all x in the range 0 ≤ x ≤ 2 = x2/8, for all x in the range 2 < x ≤ k (a) What is the value of k?(b) What is E(X)? Note: Answers to both questions are irrational numbers.arrow_forwardX and Y are random variables with the following joint PMF. X?, Y? 6 -7 9 9 0.2 0.1 0.16 -3 0.1 0.15 ??? Calculate the marginal PMF for X. What is P(X = 9) * P(X = -3)? Hint: infer the bottom right corner first.arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning