a. Determine b. Determine the expected the expected value of x value of Y
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- 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?Consider 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 EXCELUse the expected value properties to obtain the E[Y] of the following system : y = 3x + 1 , where E[X] =3
- LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.
- An investor has found that company1 have an expected return on E(X) = 4% and variance for the return equal V(X) = 0.49. Company 2 has E(Y) = 6% and variance V(Y) = 0.64. The correlation between the companies return is ρ(X,Y) = 0.3. The investor wants to invest p (0<p<1) in company1 and (1-p) in company2. The combined investment have a return: R = pX + (1-p)Y. Let p=0.4 such that R= 0.4X +0.6Y. Find the Expectation and variance of R. how are these results in comparison with X and Y separatley?Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0consider x-U(0,1). Let Y=eX and Z=X2. Find the expected value of random variable T=Y+Z
- Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Find the maximum likelihood estimator for θ in the pdf f(y; θ) = 2y/(1 − θ^2), θ ≤ y ≤ 1.Find E(R) and V (R) for a random variable R whose moment-generating function ismR(t) = e2t(1-3t2)-1
