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For CDF , first we find the pdf... Given function is not pdf...because for pdf , whole integration is 1...but integration of given function is 2/3.... So we let pdf is equal to multiple of this function....
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Probability and measure
Consider (21,F,P)-((0,1), B, A) and measurable function p: (0,1)R given by p(t)4t (1...
/ 4Given p > 1, find a function f such that l e L'(0,1) for <p...
/ 4Given p > 1, find a function f such that l e L'(0,1) for <p but fe L'(0,1]) for s p. Find another function g such that ge L'(0,1)) for r <p but g&L ([0,1]) for s > p.
Consider the measurable space ([0,],B(0) Define a set function P on this space as follows 12...
Consider the measurable space ([0,],B(0) Define a set function P on this space as follows 12 if 0 E E or 1 E E but not both if0 E E and l EE P(E)- 0 otherwise Is P a probability measure? Justify your answer
Fix an integer N>1, and consider the function f:[0,1]R defined as follows: if XE[0,1] and there...
Fix an integer N>1, and consider the function f:[0,1]R defined as follows: if XE[0,1] and there is an integer n with 1<n<N such that nxez, choose n with this property as small as possible, and set f(x) := 1/n^2; otherwise set f(x):=0. Show that f is 0 integrable, and S f.
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, :...
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
Let X ~ Unif(0,1). Find a function of X that has CDF F(x) = 1 ̶ x ̶ p for p > 0 (this is the P...
Let X ~ Unif(0,1). Find a function of X that has CDF F(x) = 1 ̶ x ̶ p for p > 0 (this is the Pareto distribution).
1 Fix an integer N > 1, and consider the function f : [0,1] - R...
1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
1. An application in probability (a) A function p(q) is a probability measure if p(x) >...
1. An application in probability (a) A function p(q) is a probability measure if p(x) > 0VT E R and (r) dx = 1. We first show that p(x):= vino exp(-) is a probability measure. (1) Compute dr. (ii) Show that were dr = 1. (ii) (1pt) Conclude that pr(I) is a probability measure. (b) A random variable x(): R + R is an integrable function that assigns a numerical value, X(I), to the outcome of an experiment, I, with...
Distributions Consider the function f(x)3+1-2-4 0334 (a) Can this function be used as a probability density function? If not, normalize it such that it can, and let that be p(x) (b) Create a CDF of y...
Distributions Consider the function f(x)3+1-2-4 0334 (a) Can this function be used as a probability density function? If not, normalize it such that it can, and let that be p(x) (b) Create a CDF of your probability density function, p(x) (c) Compute the expected value and variance of p(z) (d) What is the 90th percentile value of p()
Distributions Consider the function f(x)3+1-2-4 0334 (a) Can this function be used as a probability density function? If not, normalize it such...
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue...
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
/ 4Given p > 1, find a function f such that l e L'(0,1) for <p but fe L'(0,1]) for s p. Find another function g such that ge L'(0,1)) for r <p but g&L ([0,1]) for s > p.
Consider the measurable space ([0,],B(0) Define a set function P on this space as follows 12 if 0 E E or 1 E E but not both if0 E E and l EE P(E)- 0 otherwise Is P a probability measure? Justify your answer
Fix an integer N>1, and consider the function f:[0,1]R defined as follows: if XE[0,1] and there is an integer n with 1<n<N such that nxez, choose n with this property as small as possible, and set f(x) := 1/n^2; otherwise set f(x):=0. Show that f is 0 integrable, and S f.
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
1 Fix an integer N > 1, and consider the function f : [0,1] - R defined as follows: if 2 € (0,1) and there is an integer n with 1 <n<N such that nx € Z, choose n with this property as small as possible, and set f(x) := otherwise set f(x):= 0. Show that f is integrable, and compute Sf. (Hint: a problem from Homework Set 7 may be very useful for 0 this!)
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
1. An application in probability (a) A function p(q) is a probability measure if p(x) > 0VT E R and (r) dx = 1. We first show that p(x):= vino exp(-) is a probability measure. (1) Compute dr. (ii) Show that were dr = 1. (ii) (1pt) Conclude that pr(I) is a probability measure. (b) A random variable x(): R + R is an integrable function that assigns a numerical value, X(I), to the outcome of an experiment, I, with...
Distributions Consider the function f(x)3+1-2-4 0334 (a) Can this function be used as a probability density function? If not, normalize it such that it can, and let that be p(x) (b) Create a CDF of your probability density function, p(x) (c) Compute the expected value and variance of p(z) (d) What is the 90th percentile value of p()
Distributions Consider the function f(x)3+1-2-4 0334 (a) Can this function be used as a probability density function? If not, normalize it such...
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
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