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In Example 6c &I, suppose that X is uniformly distributed over (0, 1). If the discredited regions are determined by
Y and compute
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FIRST COURSE IN PROBABILITY (LOOSELEAF)
- 2. Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.7 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 3 Food C 1 1 1arrow_forwarda man is investigating the populaion of bear in two areas. Area 1 and Area 2. He expect the number of bear to be X and Y in area 1 and area 2 to be Poisson- distributeted. He expect the number og bear to be λ1 = 3 in area 1 and λ2 = 5 in area 2. Find P(X = 2) and P(X ≥3) and find an approximate value expression for P(X = Y)arrow_forward1. A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Ydenote the num- ber of hoses on the full-service island in use at that time. The joint pmf of Xand Yappears in the accompanying tabulation. y 1 p(x, y) .10 .04 .02 1 .08 .20 .06 .06 .14 .30 a. What is PX=1 and Y= 1)? b. Compute P(X<1 and Y< 1). c. Give a word description of the event {X÷0 and Y÷0}, and compute the probability of this event. d. Compute the marginal pmf of X and of Y. Using plx), what is P(X<1)? e. Are Xand Yindependent rv's? Explain.arrow_forward
- The geometric distributed lag model is y+= x+y(z₁+pzt-1+p²z₁-2+...) + ut, for t=1,2,... where [p]<1 and the error term ut satisfies E(utlZt Zt-1...)=0. Select all of the following statements that are correct. a. If {ut) is not an AR(1) process, we can use yt-2 as an instrument and apply 2SLS estimation and obtain consistent estimates. Ob. If {ut) is an AR(1) process, then we can use the OLS and obtain consistent estimates. O c. If {ut) is not an AR(1) process, we can use yt-1 as an instrument and apply 2SLS estimation and obtain consistent estimates. Od. we can use OLS and obtain consistent estimates in any case.arrow_forward1. Consider the data points (0, 1), (1, 1), (2, 5). (a) Find the piecewise function P(x) = that interpolates the given data points, where Spo(x), x € [0,1]. (p₁(x), x= [1,2], Po(x) = a + be, P₁(x) = c+dx², for some constants a, b, c, d to be determined. (b) Find the natural cubic spline Si(2) S(x) = Jso(x), x= [0,1]. $1(x), x= [1,2], that interpolates the given data points, where so, 81 are cubic functions within their respective intervals. Express the resulting polynomials in monomial form, that is, so (x) = ao + box + cox² +dox³, $₁(x) = a₁ + b₁x + ₁x³ + ₁x³, for some ao, bo, co, do, a1, b1, C₁, d₁. Hint: Recall the formula we derived in class for the cubic splines, 1 hi = oh [(+– z)*M + (z – zi)®M+] - * [+- z)M + (z − )Min] Xi + * [(2 - 2)f(z) + (z – zi)f(+)] for x = [xi,i+1], and i = 0,..., n-1. Solve for the values Mo, M₁, ... by setting up the appropriate system of equations, and use the formula for si to obtain the desired cubic spline.arrow_forwardThere are only two states of the world, when a person is well with probability (1-p) and ill with probability p, where (1-p)= 1/3 and p = 2/3. Consider Adam who has utility function U = (Y1, Y2,1 – p,p) = Y,"-P)Y?, where Y; is the income and i =1 is well and i = 2 is ill. When Adam is well, he earns $1000, but when he is ill, he losses $300 in health expenditures and earnings such that Y2 = $700. What is the maximum total premium that Adam is still willing-to-pay? (1-р).arrow_forward
- 1. Consider a set of data (x₁, y₁), i = 1, 2, ..., n and an intercept model given by: Yi = Bo + εi with E (₁) = 0, Var(₁) = o² and all &'s are independent and normally distributed. Derive the standard error of the point estimator Bo.arrow_forwardA consultant's salary, captured by the random variable Y = B + X comes from a deterministic base B = 78 and a random bonus X. The bonus has mean E[X] = 16 and variance V[X] =240. What is the expected value of the total compensation E[Y]?arrow_forwardA certain market has both an express checkout line and a superexpress checkout line. Let X₁ denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X₁ and X2 is as given in the accompanying table. X2 0 1 2 3 0 0.08 0.07 0.04 0.00 1 0.05 0.15 0.04 0.04 X1 2 0.05 0.04 0.10 0.06 3 0.00 0.04 0.04 0.07 4 0.00 0.02 0.05 0.06 (a) What is P(X₁ = 1, x2 = 1), that is, the probability that there is exactly one customer in each line? P(X₁ =1, X21) = (b) What is P(X₁ = X2), that is, the probability that the numbers of customers in the two lines are identical? P(X₁ = X2)= | (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X1 and X2. OA = {X₁ ≥ 2+ X₂ U X₂ ≤ 2 + X₁} = + 1 2 OA = {x₁ ≥2+ X₂ UX₂ = 2 + X₁ OA = {x₁ ≤ 2+ X₂UX ≤ 2+ X₁} Calculate the probability of this…arrow_forward
- A certain market has both an express checkout line and a super-express checkout line. Let X₁ denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X1 and X2 is as given in the accompanying table. X2 0 1 2 3 0 0.08 0.08 0.04 0.00 1 0.05 0.13 0.05 0.06 X1 2 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 4 0.00 0.01 0.05 0.06 The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout is X1 X2. Calculate the expected difference. -arrow_forwardIf X1, X2, ... , Xk have the multinomial distribution ofDefinition 8, show that the covariance of Xi and Xj is−nθiθj for i = 1, 2, ... , k, j = 1, 2, ... , k, and i Z j.arrow_forwardASIACELL LTE 5:51 PM 36% A cis.turath.edu.iq 2 of 6 Determine the Domain •B. and the Ran ge of the Fallowing fun etions 1. Fux = 6x+ 2 2 Faw = x²- 2x + 6 3. Fer)= -| + 2x - x? 4. F&) = X - 3X x²- 3x² - 5 x + 15 5. hcx)z 2K + 6 X -4 X - 12 NB. Solving s Graphing the functions are required A. Find fo llow ing Functions the inuerse of thearrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning