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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 67E

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Q10 Random variables X and Y have joint probability mass function
e-74m 3n-m
Р(X — т, Ү — п)
т 3 0, 1, 2.., п,
п 3D 0, 1, 2,...
m!(n – m)!'
Q10 (i.) Find the marginal probability mass function of X and the marginal probability mass function of Y.
Q(10(ii.) Are the random variables X and Y are independent? Justify your answer.
Q10 (iii.)
Show that the random variables X and Y – X are independent.
Q10(iv.) Deduce the p.m.f. of Y – X from part (iii).

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b) Let Z₁ =*-~N(0,1), and W₁=~N(0,1), for i=1,2,3,...,10, then: i) State, with parameter(s), the probability distribution of the statistic, T = - 10 ii) Find the mean and variance of the statistic T = iii) Calculate the probability that a statistic T = Z₁ Σας Ζα + W₁ is at most 4.
b) Let Z₁ = X-XN (0,1), and W₁ dx YHY~N(0,1), for i = 1,2,3,...,10, then: dy i) State, with parameter(s), the probability distribution of the statistic, T = - 54 1² ii) Find the mean and variance of the statistic T = Σ},wp? Σ1,2,3 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ẞ such that P(T> B) = 0.01, where T = Σ₁Z₁² + ₁ W₁².
Q4) The probability mass function of Y is f(y) = y/6 for y=1,2,3,4. Find the variance of Y.
Let joint Probability distribution function of random variables X and Y be if (x, y) = (1, 1) 1/3 |1/3 if (x, y) = (0, 0) (1/3 if (x, y) = (2, 0) P(x, y)={ otherwise . Are X and Y independent?
(a) If X1, X2, ..,Xg N(0, 52), then the distribution of is V52/8 (A) N0, V52/8 1 (В) N0, (D) N(ï, 1); (C) N(0, 1); (E) Unif[0, 1]. V52/8) (b) If B1 ~ Binom(1, p) and B2 ~ Bernoulli(p), independently of each other, then B1 + B2 follows (A) Binom(1, p); (B) Bernoulli(p); (C) Binom(2, p); (D) Binom(3, p²); (E) none of these.
Prove this
Q5 Let = {} 1 f(x) = x = 1, 2, 3, 4, 5, 6, otherwise. be the probability mass function of a discrete random variable X. Show that the probability mass function of the smallest observation Y₁ of a random sample of size 5 from this distribution is 5 91 (y₁) = = (²-")" - (0 =³) "₁ 6 6 y = 1, 2, 3, 4, 5, 6. and zero otherwise. Note: you already know how to write down the cumulative distribution function of Y₁, the smallest of X₁, X₂, X3, X₁, and X5; remembering that we are sampling from a discrete distribution, derive the probability mass function of Y₁.
Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variables
Let f(x,0)= ex-, the likelihood function is Select one: a. L(e) = e" II x8-1 i= 1 O b. L(e) = In (ex-1) C. L(e) = In (II ex- 1) d. L(e) = en Exe- 1
ExeR. 2. PrOve That opTimaL Variance V = と」 Di bouND Saris Fies THe uppeR V < min Ai
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3 Let X be a random variable with probability Law P (X = r) = q, K for r = 1,2,3.. . 00, Find the moment generating function and mean of ihe random variable X. Let K+ q = 1.

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