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please explian every thing consider two probability spaces ([0,1], β,A) and ([0,1], β,P), where Piand P2 are defi...
Question 2 0 out of 4 points Let f(x,y) Xy be defined on the rectangle R [0,1]x [0, 1] and consider the partition of R given by P2RR1.2. R2.1, R2.21. where Rx Compute U(f,P2)-L(f,P2). Please give you...
Question 2 0 out of 4 points Let f(x,y) Xy be defined on the rectangle R [0,1]x [0, 1] and consider the partition of R given by P2RR1.2. R2.1, R2.21. where Rx Compute U(f,P2)-L(f,P2). Please give your answer in decimal form. 2 '2
Question 2 0 out of 4 points Let f(x,y) Xy be defined on the rectangle R [0,1]x [0, 1] and consider the partition of R given by P2RR1.2. R2.1, R2.21. where Rx Compute U(f,P2)-L(f,P2). Please give your...
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based...
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based on the principle: rf (x) dr ES Where S is the set of values on which the function is defined i Compute the median y where: f(z) dz = Where m is the minimum value on which the function is defined.
Consider the following probability density function:...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Let p=(p1,p2,…,pm) be a probability distribution over m possible outcomes. The entropy of p is a measure of how much ran...
Let p=(p1,p2,…,pm) be a probability distribution over m possible outcomes. The entropy of p is a measure of how much randomness there is in the outcome. It is defined as where ln denotes natural logarithm. We wish to ascertain whether F(p) is a convex function of p. As usual, we begin by computing the Hessian. A) Consider the specific point p=(1/m,1/m,…,1/m). What is the (1,1) entry of the Hessian at this point? Your answer should be a function of m....
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function wit...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 oth...
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 otherwise (a) Determine the moment generating function of Y (b) Suppose the random variables X each have a continuous uniform distribution on [0,1 for i 1,2. Show that the random variable Z X1X2 has the same distribution = as the random variable Y defined above.
Consider the random variable Y, whose probability density function is defined...
Problem 2: Consider two probability density function on [0,1] : fo(x) = 1 and fi(x) =...
Problem 2: Consider two probability density function on [0,1] : fo(x) = 1 and fi(x) = 2x (a) Construct the most powerful test for Ho : X~ fo against H:Xfi with the signifi- cance level x = 0.1 (b) Find its power. (c) Suppose we observe X= 5/6. What is the p-value of the test in 2(a).
(21%) Consider the parallel circuit with probability as follows: 3. C1 with success prob p1 C2...
(21%) Consider the parallel circuit with probability as follows: 3. C1 with success prob p1 C2 with success prob p2 (a) 3%) Compute the probability p of sending data successfully through this circuit as a function of pı and p2 (b) (3%) If p1 p2 1, find the maximum ql and also the minimum q2 of p. Prove or reason that the maximum value qi of p and minimum value q2 of p you find are really the maximum and...
Please give detailed steps. Thank you. 2. Consider the following joint distribution of two discrete variables...
Please give detailed steps. Thank you.
2. Consider the following joint distribution of two discrete variables X and Y: fx,y(x, y) 01 2 3 お88 Recall that the marginal distribution of X is defined as: fx(x) and the marginal distribution of Y is defined as fy(v) -xf(i) Find fx(x) and fy(y) in the support of X and Y (or in simpler terms, find 1), P(Y = 0), P(Y-1), P(Y-2) and P(Y P(X-0), P(X 3)) b. The conditional density of Y...
1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and...
1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and g() = where ci and ca are finite real numbers. 265. for <y<02, (a) Show that f(r)dx = 9(r)dt = 1. (b) Find the cumulative distribution functions F(x) and Gu). (d) Show that if X-f(x), then 1-X g(x). (e) Show that if X h(x) = 21, for 0 <<1, then Y = c +(2-c)X ~f. (h) Show that if Uſ and U2 are two...
Question 2 0 out of 4 points Let f(x,y) Xy be defined on the rectangle R [0,1]x [0, 1] and consider the partition of R given by P2RR1.2. R2.1, R2.21. where Rx Compute U(f,P2)-L(f,P2). Please give your answer in decimal form. 2 '2
Question 2 0 out of 4 points Let f(x,y) Xy be defined on the rectangle R [0,1]x [0, 1] and consider the partition of R given by P2RR1.2. R2.1, R2.21. where Rx Compute U(f,P2)-L(f,P2). Please give your...
Consider the following probability density function: -x-1/2e-z/2 for x > 0. f(x) = the area under the curve (integral) is equal to one, then: i) Compute the mean of the function numerically based on the principle: rf (x) dr ES Where S is the set of values on which the function is defined i Compute the median y where: f(z) dz = Where m is the minimum value on which the function is defined.
Consider the following probability density function:...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 otherwise (a) Determine the moment generating function of Y (b) Suppose the random variables X each have a continuous uniform distribution on [0,1 for i 1,2. Show that the random variable Z X1X2 has the same distribution = as the random variable Y defined above.
Consider the random variable Y, whose probability density function is defined...
Problem 2: Consider two probability density function on [0,1] : fo(x) = 1 and fi(x) = 2x (a) Construct the most powerful test for Ho : X~ fo against H:Xfi with the signifi- cance level x = 0.1 (b) Find its power. (c) Suppose we observe X= 5/6. What is the p-value of the test in 2(a).
(21%) Consider the parallel circuit with probability as follows: 3. C1 with success prob p1 C2 with success prob p2 (a) 3%) Compute the probability p of sending data successfully through this circuit as a function of pı and p2 (b) (3%) If p1 p2 1, find the maximum ql and also the minimum q2 of p. Prove or reason that the maximum value qi of p and minimum value q2 of p you find are really the maximum and...
Please give detailed steps. Thank you.
2. Consider the following joint distribution of two discrete variables X and Y: fx,y(x, y) 01 2 3 お88 Recall that the marginal distribution of X is defined as: fx(x) and the marginal distribution of Y is defined as fy(v) -xf(i) Find fx(x) and fy(y) in the support of X and Y (or in simpler terms, find 1), P(Y = 0), P(Y-1), P(Y-2) and P(Y P(X-0), P(X 3)) b. The conditional density of Y...
1. Consider the following two probability density functions: f(3) = 2053 } for a <I<02 and g() = where ci and ca are finite real numbers. 265. for <y<02, (a) Show that f(r)dx = 9(r)dt = 1. (b) Find the cumulative distribution functions F(x) and Gu). (d) Show that if X-f(x), then 1-X g(x). (e) Show that if X h(x) = 21, for 0 <<1, then Y = c +(2-c)X ~f. (h) Show that if Uſ and U2 are two...
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