Why does E (x^3) =0? If X-N(0,1) I know it’s an odd function and by symmetry...
Why does E (x^3) =0? If X-N(0,1) I know it’s an odd function and by symmetry it’s 0, is there any other explanation?
Homework Answers
We know that the density of standard normal distribution is symmetric about zero, but we have standard cauchy distribution also that is symmetric about zero too, but Standard cauchy does not have any moments , so why here?
Well because normal density has negative exponential squared function that takes every polynomial multiplied to it goes to zero as we go at higher and higher values . Since normal density is sufficiently smooth , its all moments exists also , hence due to unique median as zero (standard normal case), all the integral that sums up from the left(-infinity) to 0 this just equals negative times the integral from 0 to right(infinity) for any odd function as integrand.
But anyway , symmetry is the basic cause to do so.
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Why does E (x^3) =0? If X-N(0,1) I know it’s an odd function and
by symmetry...
x2 when x E [0,1]. 1. (Total marks 12) Suppose f(x) (a) Sketch the periodic odd...
x2 when x E [0,1]. 1. (Total marks 12) Suppose f(x) (a) Sketch the periodic odd extension of this function on the interval [-3,31. You do NOT need to indicate what happens at any discontinuities. (4 marks) (b) Sketch the periodic even extension of this function on the interval -3,31. You do NOT need to indicate what happens at any discontinuities (4 marks) (c) The following graph shows f (x) along with a partial sum of the sine series for...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
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...
7. Suppose that X; ~ N(,0), i = 1,..., n and Zi ~ N(0,1), i =...
7. Suppose that X; ~ N(,0), i = 1,..., n and Zi ~ N(0,1), i = 1,..., k, and all variables are independent. State the distribution of each of the following: (a) (b) South (c) VE Vrk(x-1) 2
For number 8 I know the answer is B but I don’t know why it’s not...
For number 8 I know the answer is B but I don’t know why it’s
not A and for 18 plz draw starting material and plz show step by
step mechanism.
8. What are the missing reagents in the following reaction? I 1)? Step 1 Step 2 (a) NH3 (b) NaNH2 1-bromo-3-methylbutane (c) NH3 (d) NaNH2 1-chloro-2-methylpropane (e) NaNH2 1-iodo-4-methylpentane 1-iodo-3-methylbutane 1-bromo-4-methylpentane 18. 1) excess NaNH2, heat 2) H20 3) HgSO4/H2SO/H2O
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the...
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the sine terms, and can be approximate an odd function, so Fo(x) b sinx)+b2 sin(2x)+ b, sin(3x)+. Why is there no b, term in the series F, (x)? 1. 2. Using steps similar to those outlined for even functions, develop a rule for finding the coefficients to approximate any odd function on the interval [-π, π]. 3. If f (x)sin...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an
Suppose f is a continuous and differentiable function on...
In the following, x, (t)-Evenx(i), x,(1)-Odd{x(t): l n20 u(t)- «[n]- δ[n]-(0 otherwise δ(r) is the Dirac...
In the following, x, (t)-Evenx(i), x,(1)-Odd{x(t): l n20 u(t)- «[n]- δ[n]-(0 otherwise δ(r) is the Dirac delta function
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform?
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
2) (3 points; J2) The function below is even, odd, or nelther even nor odd. Select...
2) (3 points; J2) The function below is even, odd, or nelther even nor odd. Select the statement below which best describes which it is and how you know. f(x) = 5 x6 - x2 + 8 A) This function is even because f(-x) = f(x). B) This function is odd because f(-x) = -f(x). C) This function is even because f(-x) = -f(x). D) This function is odd because f(-x) = f(x). E) This function is neither even nor...
x2 when x E [0,1]. 1. (Total marks 12) Suppose f(x) (a) Sketch the periodic odd extension of this function on the interval [-3,31. You do NOT need to indicate what happens at any discontinuities. (4 marks) (b) Sketch the periodic even extension of this function on the interval -3,31. You do NOT need to indicate what happens at any discontinuities (4 marks) (c) The following graph shows f (x) along with a partial sum of the sine series for...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
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...
7. Suppose that X; ~ N(,0), i = 1,..., n and Zi ~ N(0,1), i = 1,..., k, and all variables are independent. State the distribution of each of the following: (a) (b) South (c) VE Vrk(x-1) 2
For number 8 I know the answer is B but I don’t know why it’s
not A and for 18 plz draw starting material and plz show step by
step mechanism.
8. What are the missing reagents in the following reaction? I 1)? Step 1 Step 2 (a) NH3 (b) NaNH2 1-bromo-3-methylbutane (c) NH3 (d) NaNH2 1-chloro-2-methylpropane (e) NaNH2 1-iodo-4-methylpentane 1-iodo-3-methylbutane 1-bromo-4-methylpentane 18. 1) excess NaNH2, heat 2) H20 3) HgSO4/H2SO/H2O
Fourier Series for Odd Functions Recall that if f is an odd function, f(-x)f(x). An odd Fourier series has only the sine terms, and can be approximate an odd function, so Fo(x) b sinx)+b2 sin(2x)+ b, sin(3x)+. Why is there no b, term in the series F, (x)? 1. 2. Using steps similar to those outlined for even functions, develop a rule for finding the coefficients to approximate any odd function on the interval [-π, π]. 3. If f (x)sin...
Suppose f is a continuous and differentiable function on [0,1] and f(0)= f(1). Let a E (0, 1). Suppose Vr,y(0,1) IF f'(x) 0 and f'(y) ±0 THEN f'(x) af'(y) Show that there is exactly f(ax) and f'(x) 0 such that f(x) one Hint: Suppose f(x) is a continuous function on [0, 1] and f(0) x € (0, 1) such that f(x) = f(ax) f(1). Let a e (0,1), there exists an
Suppose f is a continuous and differentiable function on...
In the following, x, (t)-Evenx(i), x,(1)-Odd{x(t): l n20 u(t)- «[n]- δ[n]-(0 otherwise δ(r) is the Dirac delta function
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform?
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
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