Let X and Y be two independent and identically distributed random variables that take only positive...
Let X and Y be two independent and identically distributed random variables that take only positive integer values. Their PMF is pX(n)=pY(n)=2−n for every n∈N , where N is the set of positive integers. Fix a t∈N . Find the probability P(min{X,Y}≤t) . Your answer should be a function of t . unanswered Find the probability P(X=Y) . unanswered Find the probability P(X>Y) . Hint: Use your answer to the previous part, and symmetry. unanswered Fix a positive integer k . Find the probability P(X≥kY) . Your answer should be a function of k . Hint: You may find the following facts useful. ∑∞k=j2−k=2−j+1 . For a geometric series, an=a1rn−1 , we have ∑∞n=1an=a1/(1−r) . unanswered
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Let X and Y be two independent and identically distributed
random variables that take only positive...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
Let X and Y be two independent and identically distributed random variables with expected value 1...
Let X and Y be two independent and identically distributed random variables with expected value 1 and variance 2.56. First, find a non-trivial upper bound for P(|X + Y − 2| ≥ 1). Now suppose that X and Y are independent and identically distributed N(1,2.56) random variables. What is P(|X + Y − 2| ≥ 1) exactly? Why is the upper bound first obtained so different from the exact probability obtained?
Suppose that X and Y are independent, identically distributed, geometric random variables with parameter p. Show...
Suppose that X and Y are independent, identically distributed, geometric random variables with parameter p. Show that P(X = i|X + Y = n) = 1/(n-1) , for i = 1,2,...,n-1
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables,...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
PLEASE SOLVE ONLY QUESTION B B. Let be identically and independently distributed exponential random variables with...
PLEASE SOLVE ONLY QUESTION B
B. Let
be identically and independently distributed exponential random
variables with each having probability density function
. Then, find the probability density function of
HINT- Use the following decomposition:
A. LetX1,X2, ..., Xn be identically and independently distributed random variables with each having zero mean and variance σ. If j is defined as z,-X -X, j -1,2,..n where 7t k-1 then find E(Z,) and Var Z)
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common...
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common density function f(x) 2x 0〈x〈1 0 otherwise Find the probability density function of X Y. 42.5 If 0 < a < l then ÍxHY(a) 2a3. If 1 < a < 2 then ÍxHY(a) -릎a3 + 4a-3. If a 〉 2 then fx+y(a) 0 and 0 otherwise.
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write...
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with...
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
Ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that t...
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
13. Let X1, X2, ...,Xy be a sequence of independent and identically distributed discrete random variables, each with probability mass function P(X = k)=,, for k = 0,1,2,3,.... emak (a) Find the expected value and the variance of the sample mean as = N&i=1X,. (b) Find the probability mass function of X. (c) Find an approximate pdf of X when N is very large (N −0).
PLEASE SOLVE ONLY QUESTION B
B. Let
be identically and independently distributed exponential random
variables with each having probability density function
. Then, find the probability density function of
HINT- Use the following decomposition:
A. LetX1,X2, ..., Xn be identically and independently distributed random variables with each having zero mean and variance σ. If j is defined as z,-X -X, j -1,2,..n where 7t k-1 then find E(Z,) and Var Z)
Problem 42.5 Let X and Y be two independent and identically distributed random variables with common density function f(x) 2x 0〈x〈1 0 otherwise Find the probability density function of X Y. 42.5 If 0 < a < l then ÍxHY(a) 2a3. If 1 < a < 2 then ÍxHY(a) -릎a3 + 4a-3. If a 〉 2 then fx+y(a) 0 and 0 otherwise.
Problem 4 Let X and y be independent Poisson(A) and Poisson(A2) random variables, respectively. i. Write an expression for the PMF of Z -X + Y. i.e.. pz[n] for all possible n. ii. Write an expression for the conditional PMF of X given that Z-n, i.e.. pxjz[kn for all possible k. Which random variable has the same PMF, i.e., is this PMF that of a Bernoulli, binomial, Poisson, geometric, or uniform random variable (which assumes all possible values with equal...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
Problem 9: 10 points Suppose that X, Y are two independent identically distributed random variables with the density function f(x)= λ exp (-Az), for >0. Consider T- and find its cumulative distribution function and density function.
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
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