(b) Assume that \( X \) is a geometric random variable with \( p = 0.2 \). Compute \( E[(X - 5)^2] \). This does not require a messy calculation!(i) 10(ii) 15(iii) 20(iv) 25 **Instructions**Complete the following multiple choice questions. No work will be graded in the bonus problem.**(a)** Assume that \( X \) is a random variable with \( E(X) = -3 \) and \( E(X^2) = 16 \). Compute \( E[-X(3X - 2)] \).(i) \(-54\) (ii) \(-48\) (iii) \(-33\) (iv) \(-27\)

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Answered: Complete the following multiple choice…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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1 In class, the definition of the sample variance was n-1 E-1(x; – x)². Show that i=1 1 this is equivalent to п-1 E (x;²) – Li=14)). (Hint: You can start on either side n and work to the other equation, but it might be easier to start with the original equation)
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1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are two functions. To compute the mean of Y, we can choose any of the following equivalent expressions: E(Y)= | fr (u)ydy = | fn(x)(u)g2(u)du = / fx(#)g2(9n(x)dr (1) -00 where fy (y), f91(x)(u) and fx(x) are the probability density functions (p.d.f.s) of Y, 91(X) and X, respectively. Now consider Y = X? where X is N(0, o2). a) Determine the p.d.f. fy (y) of Y. Hint: for y > 0, fy (y) Prob(Y < y) = Prob(-< X < VI) = % x (x)dx. b) Then evaluate explicitly E(Y)= S fy(y)ydy.
2. (Corrected.) Suppose Y is a random variable with P{Y = -2} = 2/10, P{Y=2} = 5/10 and P{Y = 3} = 3/10. (a) Compute the mean of Y. (b) Compute E[Y²]. (c) Compute the variance of Y. (d) Compute E[Y+]. (e) Compute E[|Y|]. (f) Compute E[Y^2].
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 ?x and ?y, variances ?2x and ?2y, and population correlation coefficient ? (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 ?x?y? is called the covariance of x and y. If x and y are independent, then ? = 0 and the formula for ?2w 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 fund…
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2. Let X and Y be independent random variables with E[X] = 4, Var[X] = 84, E[Y] = 6, and Var[Y] = 54. Find E[5X + 2Y - 18] - b. Find E[-3Y+X+3] -a. -с. Find E[X2] d. Find ox e. Find E[Y2] f. Find Oy g. Find Var[X+3Y - 6] h. Find Var[3X - 2Y + 35] Find E[(2X - 8)/5] j. Find Var[(2X - 8)/5] i.
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Please solve using R

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Complete the following multiple choice questions. No work will be graded in the bonus problem. (а) E(X²) = 16. Compute E[-X(3X – 2)]. Assume that X is a random variable with E(X) = -3 and (i) -54 (ii) -48 (iii) 33 (iv) -27