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Suppose n numbers ave vandoml Selected without vepacement hom a set ot N numbers, that is,...
Problem 2. Let n be a positive integer. We sample n numbers ai,...,an from the set...
Problem 2. Let n be a positive integer. We sample n numbers ai,...,an from the set 1, 2,...,n} uniformly at random, with replacement. Say that the picks i and j with i < j are a match if a -aj. What is the expected total number of matches? Hint: Use indicators. Wİ
Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let...
Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let Y be the random variable representing the smallest of the numbers. (a) Show that the probability density of Y is given by py (t) -61-t)5, 0t <1 [51 Hint: The probability density for the r-th largest random variable can be derived using the Beta distribution by letting a = r and ?-n-r +1. (b) What is the probability that the smallest number is less...
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,...
Suppose that n students are selected at random without replacement from a class containing 28 students,...
Suppose that n students are selected at random without replacement from a class containing 28 students, of whom 8 are boys and 20 are girls. We assume that 0 < n < 28. Let X denote the number of boys that are obtained. Answer the following questions: a (4 marks) State the distribution of X, with parameters b (1 mark) Write down the possible values of X c (1 mark) Express E(X) in terms of n. d (4 marks) For...
Problem 1: Dynamic Programming in DAG Let G(V,E), |V| = n, be a directed acyclic graph...
Problem 1: Dynamic Programming in DAG Let G(V,E), |V| = n, be a directed acyclic graph presented in adjacency list representation, where the vertices are labelled with numbers in the set {1, . . . , n}, and where (i, j) is and edge inplies i < j. Suppose also that each vertex has a positive value vi, 1 ≤ i ≤ n. Define the value of a path as the sum of the values of the vertices belonging to...
1. An individual who has car insurance from a certain company is randomly selected. Let Y...
1. An individual who has car insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The probability distribution of Y is given in the table below: Ply) 0. 600 .25 0.10 0.05 a. Determine the expected value of Y, that is E(Y). [4 points) E(Y)=0(0.6) + 110.25) +2(0.10) +3(0.05) E(Y)=0.6 b. Suppose an individual with Y violations incurs a surcharge of $100XY?....
Suppose that X = (Xi, X2, , X.) and Y-X,,Y2, , Ym) are random samples from...
Suppose that X = (Xi, X2, , X.) and Y-X,,Y2, , Ym) are random samples from continuous distributions F and G, respectively.Wilcoxon's two-sample test statistic W - W(X, Y) is defined to beRi where R, is the rank of Y in the combined sample 1. Let T Σǐn i Zi where Zi,Z2, ,Zm are numbers sampled at random without replacement from the set {1,2,..., N} Show that E(Z) = (N + 1)/2 and hence E(T) m(N + 1)/2 Show that...
i need help with 2b please is a set of input values, Y- 2. In this question, we reuse the notation of lecture 37: X-{xi, ,x , m-1) is a set of hash values, and H is an [X → Y)-valued random variab...
i need help with 2b please
is a set of input values, Y- 2. In this question, we reuse the notation of lecture 37: X-{xi, ,x , m-1) is a set of hash values, and H is an [X → Y)-valued random variable {0.1, In lecture, we showed that for any hash value y e Y, the expected number of input values that hash to y is k/m, where k XI and m Yl. However, in determining the time it...
Suppose that Case i is an outlier in the sense that an amount δ is added to its expected value. Let u i be the unit vector with 1 in the ith position and zeros elsewhere and write the revised model is...
Suppose
that Case i is an outlier in the sense that an amount δ is added to
its expected value. Let u i be the unit vector with 1 in the ith
position and zeros elsewhere and write the revised model is y = Xβ
+ δu i + e. Weisberg (1985) calls this the mean shift outlier
model. Using standard methodology, develop the test statistic for
the hypothesis, H 0 : δ = 0 and show that the resulting...
Suppose there are 40 students in my class. Out of which, 8 students live on campus...
Suppose there are 40 students in my class. Out of which, 8 students live on campus housing. If I select 15 students at random from this class of 40, find the probability that more than 3 live on campus housing. (Bonus) 35.18% 39.80% 64.81% 25.01% 60.19% A box contains 4 red and 6 black balls if two balls are selected at random without replacement and colors of the balls are noted. In this experiment, we don't know how many red...
Problem 2. Let n be a positive integer. We sample n numbers ai,...,an from the set 1, 2,...,n} uniformly at random, with replacement. Say that the picks i and j with i < j are a match if a -aj. What is the expected total number of matches? Hint: Use indicators. Wİ
Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let Y be the random variable representing the smallest of the numbers. (a) Show that the probability density of Y is given by py (t) -61-t)5, 0t <1 [51 Hint: The probability density for the r-th largest random variable can be derived using the Beta distribution by letting a = r and ?-n-r +1. (b) What is the probability that the smallest number is less...
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,...
Suppose that n students are selected at random without replacement from a class containing 28 students, of whom 8 are boys and 20 are girls. We assume that 0 < n < 28. Let X denote the number of boys that are obtained. Answer the following questions: a (4 marks) State the distribution of X, with parameters b (1 mark) Write down the possible values of X c (1 mark) Express E(X) in terms of n. d (4 marks) For...
1. An individual who has car insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The probability distribution of Y is given in the table below: Ply) 0. 600 .25 0.10 0.05 a. Determine the expected value of Y, that is E(Y). [4 points) E(Y)=0(0.6) + 110.25) +2(0.10) +3(0.05) E(Y)=0.6 b. Suppose an individual with Y violations incurs a surcharge of $100XY?....
Suppose that X = (Xi, X2, , X.) and Y-X,,Y2, , Ym) are random samples from continuous distributions F and G, respectively.Wilcoxon's two-sample test statistic W - W(X, Y) is defined to beRi where R, is the rank of Y in the combined sample 1. Let T Σǐn i Zi where Zi,Z2, ,Zm are numbers sampled at random without replacement from the set {1,2,..., N} Show that E(Z) = (N + 1)/2 and hence E(T) m(N + 1)/2 Show that...
i need help with 2b please
is a set of input values, Y- 2. In this question, we reuse the notation of lecture 37: X-{xi, ,x , m-1) is a set of hash values, and H is an [X → Y)-valued random variable {0.1, In lecture, we showed that for any hash value y e Y, the expected number of input values that hash to y is k/m, where k XI and m Yl. However, in determining the time it...
Suppose
that Case i is an outlier in the sense that an amount δ is added to
its expected value. Let u i be the unit vector with 1 in the ith
position and zeros elsewhere and write the revised model is y = Xβ
+ δu i + e. Weisberg (1985) calls this the mean shift outlier
model. Using standard methodology, develop the test statistic for
the hypothesis, H 0 : δ = 0 and show that the resulting...
Suppose there are 40 students in my class. Out of which, 8 students live on campus housing. If I select 15 students at random from this class of 40, find the probability that more than 3 live on campus housing. (Bonus) 35.18% 39.80% 64.81% 25.01% 60.19% A box contains 4 red and 6 black balls if two balls are selected at random without replacement and colors of the balls are noted. In this experiment, we don't know how many red...
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