Question 5. Let r, n be positive integers. 1. (6 pts) Consider the random binary r...
Question 5. Let r, n be positive integers.
1. (6 pts) Consider the random binary r n matrix M , where each entry is equal to 0 or 1 with probability 1/2 (so each entry follows the Bernoulli random variable with parameter 1/2), and these entries are (jointly) independent random variables. What is the probability that each column in M has at most one entry with 1?
1
2. (*, 4 pts) Let S1, . . . , Srbe identically and independently distributed random subsets of
{1, . . . , n}, where S1 is equal to each subset of {1, . . . , n} with equal probability (i.e. 1/2n). What is the probability that S1, . . . , Srare disjoint? Justify your answer.
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Question 5. Let r, n be positive
integers.
1. (6 pts) Consider the random binary r...
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)
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 ...
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 , wherepEPr(X:-)-0.7 and Pr(X, =0).1-p=0.3. a) (10 pts.) Complete the table where k denotes the number of 1's in the n! sequence, andkkn-k b) (10 pts.) Calculate H(X) c) (10 pts.) Assume that Pr[T)]21-ε 0.9. Find the corresponding typical sequence set n) d) (10 pts.) Assume Pr[ 21-820.9. Find the corresponding smallest set B ). 2. Consider a random walk random variable X, on the graph in Figure 1....
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S):...
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S): the power set of a set S; namely, the set of all subsets of S. P*(S): = P(S) - {@}; namely, the set of all non-empty subsets of S. The following question is a challenging one! As a start, may be you try this question for small values of n, say n=1,2,3. Can you make a guess? (1) We all know that P*(On) has...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C C B uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X - Y). Note: X,Y can take value 0 if you pick the empty...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C CB uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X – Y). Note: X,Y can take value 0 if you pick the empty set....
1) [6 pts] Let Y be a Bernoulli random variable with success probability Pr (Y 1...
1) [6 pts] Let Y be a Bernoulli random variable with success probability Pr (Y 1 )p, and let Y, Yn be iid draws from this distribution. Let p be the fraction of successes (1's) in this sample. (a) Show that p Y. (b) Show that p is an unbiased estimator of p. (c) (1-p)/n Show that var (p)-p
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
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...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why WR Two subspaces are the same when one subspace is a subset of the other subspace. Two subspaces are the same when they are spanned by the same vectors Two subspaces are the same when they are subsets of the same space Two subspaces are the same when they have the same dimension Incorrect 0/1 pts Question 10 Let U...
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2)...
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
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)
1. Consider sequence of independent identically distributed binary random variable x,,x,,x,,x,-4 , wherepEPr(X:-)-0.7 and Pr(X, =0).1-p=0.3. a) (10 pts.) Complete the table where k denotes the number of 1's in the n! sequence, andkkn-k b) (10 pts.) Calculate H(X) c) (10 pts.) Assume that Pr[T)]21-ε 0.9. Find the corresponding typical sequence set n) d) (10 pts.) Assume Pr[ 21-820.9. Find the corresponding smallest set B ). 2. Consider a random walk random variable X, on the graph in Figure 1....
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S): the power set of a set S; namely, the set of all subsets of S. P*(S): = P(S) - {@}; namely, the set of all non-empty subsets of S. The following question is a challenging one! As a start, may be you try this question for small values of n, say n=1,2,3. Can you make a guess? (1) We all know that P*(On) has...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C C B uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X - Y). Note: X,Y can take value 0 if you pick the empty...
Problem 3. (5 pts) Discrete Random Variables (a) (3 pts) Let A = {1,2,3,4}. Pick a subset B C A uniformly among the 24 subsets (i.e. the power set of A) and let X be its size. Then likewise pick a subset C CB uniformly from the power set of B and let y be its size. Give the joint p.m.f of (X,Y) and compute E(X – Y). Note: X,Y can take value 0 if you pick the empty set....
1) [6 pts] Let Y be a Bernoulli random variable with success probability Pr (Y 1 )p, and let Y, Yn be iid draws from this distribution. Let p be the fraction of successes (1's) in this sample. (a) Show that p Y. (b) Show that p is an unbiased estimator of p. (c) (1-p)/n Show that var (p)-p
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
0/1 pts Inooreat Question 9 Suppose W is a subspace of R" spanned by n nonzero orthogonal vectors. Explain why WR Two subspaces are the same when one subspace is a subset of the other subspace. Two subspaces are the same when they are spanned by the same vectors Two subspaces are the same when they are subsets of the same space Two subspaces are the same when they have the same dimension Incorrect 0/1 pts Question 10 Let U...
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
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