I use definition of differentiation to prove this
p.v. is the cauchy principal value of in this case (1/x) д 5, Prove that a...
evaluate the principal cauchy value of the following
integrals
Determine the nature of the singularities of the following
functions
- da Sol: T (x²+4)² 00 16 13) 14) 2x ?! da Sol. I a4+5x²+4 2 8 - 15) da a²_60+25 - 00 -م : (ع) (6) 22. sn (4) -ام .ع : (ع ) ۴ (5) 22
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
(5) Let (N, A, д) be a complete measure spaсе (i.e. В € А, д(В) — 0, А с В — АЄ A, uA 0. Let f,g : 2 -> R* be a pair functions. Assume that f is measurable and that f g almost everywhere (a) Prove that g is measurable. (b) Let A E A and assume that f is integrable on A. Prove that g is integrable on A and f du g ар. A A
(5)...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
VII (5) (a) Prove the Cauchy-Schwarz inequality for vectors in R”: v•w |v||w| for all v, w ER. Also show that equality holds if and only if v = lw for some > 0. HINT: Assume, without loss of generality, that v, w # 0. Consider the non-negative function o(t) = \v – tw|2. Show that º attains a minimum at t = 6:12. Evaluate o at this point and use the fact that ¢ is non-negative to conclude. Address...
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
The standard Cauchy distribution has cumulative
distribution function
F(x) = 1 + 1 tan−1(x) 2π
where −∞ < x < ∞.
1
a. Find the probability density function of X.
b. FindxsuchthatP(X>x)=0.2.
Question 7: The standard Cauchy distribution has cumulative distribution function F(x) = + tan-1 (x) π where-00< 00. We were unable to transcribe this image
Prove that the series expansion of the exponential function is
cauchy
1 e" -Σ n! 321 x 1972-0
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First,...