Problem 28. Suppose X is a finite set of real numbers. Show that if a ≤...
Problem 28. Suppose X is a finite set of real numbers. Show that if a ≤ b then 1. P(a < X ≤ b) = F(b) − F(a). 2. P(a ≤ X ≤ b) = F(b) − F(a) + f(a). 3. P(a < X < b) = F(b) − F(a) − f(b). 4. P(a ≤ X < b) = F(b) − F(a) + f(a) − f(b). (Use 1 when showing 2-4.) Theorem 35. Every CDF function is non-decreasing with a max value of 1.
Problem 29. Explain why Theorem 35 is true.
Problem 30. If you had to guess at the minimum value of a CDF, what might you say and why?
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Problem 28. Suppose X is a finite set of real numbers. Show that
if a ≤...
Let X be a finite set and F a family of subsets of X such that...
Let X be a finite set and F a family of subsets of X such that every element of X appears in at least one subset in F. We say that a subset C of F is a set cover for X if X =U SEC S (that is, the union of the sets in C is X). The cardinality of a set cover C is the number of elements in C. (Note that an element of C is a...
4. Let A be m n and B be m x 1 . Define f : IR"-> R by (a) Quote a previous problem to show that f has a minimum. Say that the minimum (b) Find Df. (Hint: Chain Rule using the function N from...
4. Let A be m n and B be m x 1 . Define f : IR"-> R by (a) Quote a previous problem to show that f has a minimum. Say that the minimum (b) Find Df. (Hint: Chain Rule using the function N from Problem 67.) occurs at y E R". (Note: it may be that A. y might be inconsistent.) B, since the equation A. X B (c) Apply the Interior Extreme Theorem to get an equation...
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.)...
where
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
1. Show that if X(t) is a real random process with finite mean power and a...
1. Show that if X(t) is a real random process with finite mean power and a mean power spectral density function Sx(S), then for all f. (a) Sx(f) > 0: (b) Sx(-f)=Sx(f). Hint: Recall that if x(t) is a real signal with spectrum X(f), then X (f) = X(-).
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function...
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an "unconstrained" ap- proach (i.e., variable substitution in "primal" problem)? (b) Do optimal solutions for all goods satisfy "MRS" "price ratio" condition (i.e., MRSy(c) for all (V) i j)? If so, explain why. If...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use θMLE-max Xi-X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead he'll say the second largest number: θ-Xn-1. Determine the bias of this estimator by carefully finding the density function...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use alLE = max Xi = X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead, he'll say the second largest number: θ = Xn-1. Determine the bias of this estimator...
Question 1: Let R be the set of real numbers and let 2R be the set...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Theorem 2.2: The values F(x) of the distribution function of a diserete random variable X satisfy...
Theorem 2.2: The values F(x) of the distribution function of a diserete random variable X satisfy the conditions 1) F)-0 and F()-1 2) If a s b, then Fla) S F(b) for any real numbers a and b Example 5: Find the edf of X for example 3. F(1) PX-1) 2 F(2) P(x s 2) -Px)P2020 F(5)-P(X55) = P(x-1)+P(X-2) + p(X-3) + P(X-4) + P(X-5)-1 raph of the cdf for example 5 will looks like cdf 14/20 9/20 5/20 2/20
233 Theorem. (1) If 31,23,. iTn are linearly independent vectors in X then there are TA -İ, in X"...
prove (3)
233 Theorem. (1) If 31,23,. iTn are linearly independent vectors in X then there are TA -İ, in X" such, that' A(x)=6',ond (2) If X is infinite dimensional then so is X 3) Every finite dimensional vector subspace of X has a complement. (4) If Y is a finite dimensional vector subspace of X then Y = ran P for some bounde idempotent linear map P:X X Prof. (1) Let := span xi. Then Y, is a closed...
4. Let A be m n and B be m x 1 . Define f : IR"-> R by (a) Quote a previous problem to show that f has a minimum. Say that the minimum (b) Find Df. (Hint: Chain Rule using the function N from Problem 67.) occurs at y E R". (Note: it may be that A. y might be inconsistent.) B, since the equation A. X B (c) Apply the Interior Extreme Theorem to get an equation...
where
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
1. Show that if X(t) is a real random process with finite mean power and a mean power spectral density function Sx(S), then for all f. (a) Sx(f) > 0: (b) Sx(-f)=Sx(f). Hint: Recall that if x(t) is a real signal with spectrum X(f), then X (f) = X(-).
2. Consider the utility maximization problem with n goods (a finite) (a) If the utility function u(c) is strictly concave, increasing, C1, and as- suming interiority of the optimal solution, what is the problem the consumer is solving? What are the FOCs for this problem using an "unconstrained" ap- proach (i.e., variable substitution in "primal" problem)? (b) Do optimal solutions for all goods satisfy "MRS" "price ratio" condition (i.e., MRSy(c) for all (V) i j)? If so, explain why. If...
2. Suppose you decide to randomly generate numbers from X ~ Unif (0,0). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use θMLE-max Xi-X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead he'll say the second largest number: θ-Xn-1. Determine the bias of this estimator by carefully finding the density function...
2. Suppose you decide to randomly generate numbers from X ~ Unif(0, ). Your friend will ask for n numbers and then use this information to guess what value you (secretly) chose for θ. Typically, one might use alLE = max Xi = X, to estimate θ. Your friend, however, has meganumerophobia, and is afraid to say the maximum number in the random sample. Instead, he'll say the second largest number: θ = Xn-1. Determine the bias of this estimator...
Question 1: Let R be the set of real numbers and let 2R be the set of all subsets of the real numbers. Prove that 2 cannot be in one-to-one correspondence with R. Proof: Suppose 2 is in one-to-one correspondence with R. Then by definition of one- to-one correspondence there is a 1-to-1 and onto function B:R 2. Therefore, for each x in R, ?(x) is a function from R to {0, 1]. Moreover, since ? is onto, for every...
Theorem 2.2: The values F(x) of the distribution function of a diserete random variable X satisfy the conditions 1) F)-0 and F()-1 2) If a s b, then Fla) S F(b) for any real numbers a and b Example 5: Find the edf of X for example 3. F(1) PX-1) 2 F(2) P(x s 2) -Px)P2020 F(5)-P(X55) = P(x-1)+P(X-2) + p(X-3) + P(X-4) + P(X-5)-1 raph of the cdf for example 5 will looks like cdf 14/20 9/20 5/20 2/20
prove (3)
233 Theorem. (1) If 31,23,. iTn are linearly independent vectors in X then there are TA -İ, in X" such, that' A(x)=6',ond (2) If X is infinite dimensional then so is X 3) Every finite dimensional vector subspace of X has a complement. (4) If Y is a finite dimensional vector subspace of X then Y = ran P for some bounde idempotent linear map P:X X Prof. (1) Let := span xi. Then Y, is a closed...
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