Exercise 2. Let p E [0, 1] and let random variables X1,..., X, be i.i.d., such that P(X; = 1) = pand P(X; = 0) = 1 – p for all i E {1, 2, . . . ,n}. Finally, let X be the average of the X;.a) Determine E(X;) and Var(X;).b) Determine E(X) and Var(X).c) Show thatP(X = k/n) = (") p*(1 - p)"-.k

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Answered: Exercise 2. Let p E [0, 1] and let…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Let JO, J1,..., J4 independent random variables according to the Ber (r;) law, where i = 0, 1,..., 4, respectively. We define the random variables Xi = min {JO + Ji, 1}, for i = 1, 2, 3, (a) Find the law of Xi , for each i = 1, 2, 3, 4. (b) Find the law of (X1, X2, X3, X4).
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For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. (d) X = the length of a randomly selected rattlesnake Describe the set of possible values for the variable. 0, 1, 2, 3, ... 1, 2, 3, ... [0, ∞) (0, ∞) State whether the variable is discrete. discrete not discrete (e) Z = the sales tax percentage for a randomly selected amazon.com purchase, where n = the total number of possible sales tax percentages for online purchases Describe the set of possible values for the variable. z1, z2, z3, ..., zn [0, zn] 1, 2, 3, ..., n [0, n] State whether the variable is discrete. discrete not discrete (f) Y = the pH of a randomly chosen soil sample Describe the set of possible values for the variable. 0, 1, 2, 3, ..., 14 [0, 14] 1, 2, 3, ..., 14 (0, 14] State whether the variable is discrete. discrete not discrete (g) X = the tension (psi) at which a randomly selected…
Prove that the events Es, E2... En are independent iff their corresponding indicator variates I,, I₂.. In are independent.
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).
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Exercise 2. Let p E [0, 1] and let random variables X1,..., Xn be i.i.d., such that P(X; = 1) = p and P(X; = 0) = 1 – p for all i E {1,2, ..., n}. Finally, let X be the average of the X;. a) Determine E(X;) and Var(X;). b) Determine E(X) and Var(X). c) Show that P(X = k/n) = )p* (1 – p)*-k.