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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 (10th 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_forwardA scientist has measured quantities y, x1,and x,. She believes that y is related to x, and x2 through the equation y = ae+Px2 8, where & is a random error that is always positive. Find a transformation of the data that will enable her to use a linear model to estimate B, and B,.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_forward
- 2. Let Y = aX + b %3D (1) Find the covariance of X and Yarrow_forwardb. Suppose that e, is zero mean white noise with var(et) = o. Consider the process: i. ii. iii. iv. Y₁ = 1+0.4Y-1 + et - 0.3e-1 0.15€t-2 Write the model using lag operator notation. Assess if the process is covariance stationary. Identify this model as an ARIMA (p, d, q) process; that is, specify p, d, and q. Find μ = E(Y).arrow_forwardConsider a probit model with an interaction term: Pr(y = 1|r) = (Bo + B1r1+ B2r2 + Bza1 x r2) a. What is marginal effect of r1? b. Now suppose B = (1, -2,3,-1) find the marginal effect of r if r1 = 0 and %3D %3D 22 = 0. c. Now suppose 3 = (1, –2, 3,-1) find the marginal effect of r1 if r1 = 0 and %3D 12 = 1. d. Now suppose B = (1,-2, 3, –1) find the marginal effect of r if r, = 1 and %3D %3D I2 = 0. e. Now suppose B = (1,-2, 3,-1) find the marginal effect of ai if ri T2 = 1. 1 andarrow_forward
- 2. 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_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. X₁ 0 1 2 3 4 X₂ 1 2 3 0 0.09 0.06 0.04 0.00 0.06 0.15 0.05 0.04 0.05 0.04 0.10 0.06 0.00 0.03 0.04 0.07 0.00 0.01 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) = 0.15 (b) What is P(X₂ = X₂), that is, the probability that the numbers of customers in the two lines are identical? P(X₂=X₂)=[ X (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 = {X₂₁ ≤ 2 + X₂ UX₂ ≥ 2 + X₂} 1 O A = {X₂₁ ≥ 2 + X₂ UX₂ ≥ 2 + X₂} 1 O A = {X₂ ≤ 2 + X₂ UX₂ ≤2+X₂} |OA = {X₁₂ ≥ 2 + X₂ UX₂ ≤2+…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. X1 0 1 2 3 4 X2 3 0 1 2 0.09 0.07 0.04 0.00 0.05 0.15 0.05 0.04 0.05 0.04 0.10 0.06 0.00 0.03 0.04 0.07 0.00 0.02 0.05 0.05 (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. O A= {X₁ ≥ 2 + X₂ U X₂2 ≥ 2 + X₁} O A = {X₁ ≤ 2 + X₂ U X₂ ≤ 2 + X₁} O A = {X₁ ≥ 2 + X₂ U X₂ ≤ 2 + X₁} O A = {X₁ ≤ 2 + X₂ U X₂ ≥ 2 + X₁}…arrow_forward2. Suppose you are to fit a linear model which is forced to have an intercept equal to 5, of the form given by: Y₁ = 5 + B₁X₁ + &; for i = 1,2,.., n where & has mean 0 and variance o² for any value of X. a) Derive the least squares estimator, ₁ of the slope ₁. b) Find E(₁) and o² (B₁). c) Derive the maximum likelihood estimator, ₁ of the slope ₁. Is it the same as the least squares estimator? d) What is the estimator of the variance of the random error term? e) Prove whether or not it is still true that Σ₁ e₁ = 0.arrow_forwardConstruct a model for the number of cats, y, after x months that make use of the following assumptions: 1. It begins with two cats – one female and one male, both unneutered. 2. Each litter is composed of 4 kittens – 3 males and 1 female. 3. It takes four months before a new generation of cats is born. 4. No cat dies (all are healthy) and no new cats are introduced.arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning