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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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A First Course In Probability, Global Edition
- 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_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_forwardJust 4.3arrow_forward
- There 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_forwardConsider the following LP model. Minimize Z= 6 x, + 3 x, + 4 x, +x, 2 x, - x, +x, - 6x, = 60 x, +x, + 2 x, 0 Convert the above LP model into standard form.arrow_forward1. Show that y₁= ez and y2 = ze form a linearly independent set on the interval (-00,00).arrow_forward
- A 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_forwardA 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_forward2. Let preferences of both individuals be given by log(c) + log(c). Suppose that the endowment vectors are wA = (5, 10) and wB clearing price and the equilibrium consumption bundles of each individual. (10, 5). Solve for the marketarrow_forward
- A 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. X1 0 1 2 3 4 b. 0 .08 .06 .05 .00 .00 1 .07 .15 .04 .03 .01 X2 2 .04 .05 .10 .04 .05 3 .00 .04 .06 .07 .06 a. What is P(X₁ = 2 and X₂ = 2), that is the probability that there is exactly one customer in each line? What is P(X₁ = X₂), that is, the probability that the numbers of customers in the two lines are identical? 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, and calculate the probability of event A, that is P(A). d. Determine the marginal pmf of X₁, and then calculate the expected number of customers in line at the express checkout. e. Determine the…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 x1 0 0.09 0.07 0.04 0.00 1 0.05 0.15 0.05 0.04 2 0.05 0.03 0.10 0.06 3 0.01 0.02 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, X2 = 1) = (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. A = {X1 ≥ 2…arrow_forwardThe following information regarding a dependent variable y and an independent variable x is provided: Σx = 90 Σ(y - ȳ)(x - x̄) = -156 Σy = 340 Σ(x - x̄)2 = 234 n = 4 Σ(y - ȳ)2 = 1974 SSR = 104 The y-intercept is a. 100. b. -100. c. -.667. d. .667arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning