2. Suppose that X is a random variable with E(X) = 6 and Var(X) = 4. Find the following: a) Е(4X + 1) b) Var(3X-2) с) Е(X). %3D
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- 1.Suppose that one word is to be selected at random from the sentence "THE GIRL PUT ON HER BEAUTIFUL RED HAT". If X denotes the number of letters in the word that is selected, (1) What is the value of E(X)? (2) What is the value of Var(X)?Suppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?A4. If random variables X and Y each have variance o? = 3 and corr(X,Y) = 0.4, what is the value of Cov(3X – 1, 2Y + 2)?
- Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.Suppose X and Y are independent random variables such that E(X) = 0, -1, Var(X)= 1, and Var(Y) = 4. Find the following: a. E(7X + 2Y – 3) b. Е(X?) с. Var(2X — 5Y + 7)2. Let the independent random variables X1 and X2 have Bin(0.1,2) and Bin(0.5, 3), respectively. (a) Find P(X1 = 2 and X2 = 2). (b) Find P(X1 + X2 = 1). (c) Find E(X1 + X2). (d) Find Var(X1 + X2).
- Suppose 2 docoss A and B test all into patients coming for syphilis. Let At ={ doctor Á makes a a clinic pesitive diagnosis} and Bt = { docbr B makes a postive diagnosis} %3D Suppose doctor A A diagnoses 10%% of all patients as postive, positive and both as docfor B diaanoser 17% of all patients2. Previously, you studied linear combinations of independent random variables. What happens if the variables are not independent? A lot of mathematics can be used to prove the following: Let x and y be random variables with means μx and μy, variances σ2x and σ2y, and population correlation coefficient ρ (the Greek letter rho). Let a and b be any constants and let w = ax + by for the following formula. μw = aμx + bμyσ2w = a2σ2x + b2σ2y + 2abσxσyρ In this formula, r is the population correlation coefficient, theoretically computed using the population of all (x, y) data pairs. The expression σxσyρ is called the covariance of x and y. If x and y are independent, then ρ = 0 and the formula for σ2w reduces to the appropriate formula for independent variables. In most real-world applications the population parameters are not known, so we use sample estimates with the understanding that our conclusions are also estimates.Do you have to be rich to invest in bonds and real estate? No, mutual…Q1. Given B = {vi = (0, 1, 1, 1), v2 = (2, 1, –1, -1), v3 = (1, 4, –1, 2), v4 = (6, 9, 4, 2)} %3! %3D B' = {wi = (0, 8, 8), w2 = (-7,8, 1), w3 = (-6,9, 1)} -2 1 6 2 1 -3 0 7 1 A = 1 and T : R4 → R3 such that matrix A is the transformation matrix in relation to B and B' basis. a)Verify that set B is a basis of R*and that the set B' is a basis of R³. b)Find [T(v1)]B, [T(v2)]B [T(v3)]B, [T(v4)]B c) find T(v1), T(v2), T(v3), T(v4)