Why is the expectation of cauchy distribution not defined? (What is the intuition behind it?)











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Let $X$ be random variable with pdf $f_X(x) = dfrac{1}{pi(1+x^2)}$. I understand that mathematically, the improper integral, $displaystyleintlimits_{-infty}^{infty}dfrac{x}{pi(1+x^2)}dx$ does not exist ($underset{T_1to-infty}{lim}underset{T_2toinfty}{lim}displaystyleintlimits_{T_1}^{T_2}dfrac{x}{pi(1+x^2)}dx = frac{ln(1+T_2^2) - ln(1+T_1^2)}{2pi} = infty - infty) implies$ undefined. However I am unable to understand the intuition, why $E[X] ne 0$ since the pdf is an even function which takes positive and negative values with equal probability. Hence large enough samples should produce mean close to 0. Please help?










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    According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
    – Jack D'Aurizio
    Nov 19 at 16:02












  • It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
    – NCh
    Nov 20 at 3:25

















up vote
0
down vote

favorite
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Let $X$ be random variable with pdf $f_X(x) = dfrac{1}{pi(1+x^2)}$. I understand that mathematically, the improper integral, $displaystyleintlimits_{-infty}^{infty}dfrac{x}{pi(1+x^2)}dx$ does not exist ($underset{T_1to-infty}{lim}underset{T_2toinfty}{lim}displaystyleintlimits_{T_1}^{T_2}dfrac{x}{pi(1+x^2)}dx = frac{ln(1+T_2^2) - ln(1+T_1^2)}{2pi} = infty - infty) implies$ undefined. However I am unable to understand the intuition, why $E[X] ne 0$ since the pdf is an even function which takes positive and negative values with equal probability. Hence large enough samples should produce mean close to 0. Please help?










share|cite|improve this question




















  • 2




    According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
    – Jack D'Aurizio
    Nov 19 at 16:02












  • It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
    – NCh
    Nov 20 at 3:25















up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Let $X$ be random variable with pdf $f_X(x) = dfrac{1}{pi(1+x^2)}$. I understand that mathematically, the improper integral, $displaystyleintlimits_{-infty}^{infty}dfrac{x}{pi(1+x^2)}dx$ does not exist ($underset{T_1to-infty}{lim}underset{T_2toinfty}{lim}displaystyleintlimits_{T_1}^{T_2}dfrac{x}{pi(1+x^2)}dx = frac{ln(1+T_2^2) - ln(1+T_1^2)}{2pi} = infty - infty) implies$ undefined. However I am unable to understand the intuition, why $E[X] ne 0$ since the pdf is an even function which takes positive and negative values with equal probability. Hence large enough samples should produce mean close to 0. Please help?










share|cite|improve this question















Let $X$ be random variable with pdf $f_X(x) = dfrac{1}{pi(1+x^2)}$. I understand that mathematically, the improper integral, $displaystyleintlimits_{-infty}^{infty}dfrac{x}{pi(1+x^2)}dx$ does not exist ($underset{T_1to-infty}{lim}underset{T_2toinfty}{lim}displaystyleintlimits_{T_1}^{T_2}dfrac{x}{pi(1+x^2)}dx = frac{ln(1+T_2^2) - ln(1+T_1^2)}{2pi} = infty - infty) implies$ undefined. However I am unable to understand the intuition, why $E[X] ne 0$ since the pdf is an even function which takes positive and negative values with equal probability. Hence large enough samples should produce mean close to 0. Please help?







probability-distributions improper-integrals means






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edited Nov 20 at 10:27

























asked Nov 19 at 15:54









sh10

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  • 2




    According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
    – Jack D'Aurizio
    Nov 19 at 16:02












  • It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
    – NCh
    Nov 20 at 3:25
















  • 2




    According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
    – Jack D'Aurizio
    Nov 19 at 16:02












  • It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
    – NCh
    Nov 20 at 3:25










2




2




According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
– Jack D'Aurizio
Nov 19 at 16:02






According to your logic, large enough samples should agree with the central limit theorem, too. But that is not the case for the Cauchy distribution: $frac{x}{x^2+1}notin L^1(mathbb{R})$, full stop.
– Jack D'Aurizio
Nov 19 at 16:02














It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
– NCh
Nov 20 at 3:25






It is well-known fact, that for Cauchy distribution, arithmetic mean of independent samples $frac{X_1+ldots+X_n}{n}$ is also Cauchy distributed with the same pdf as of summands. So, the mean of large enough samples is not close to zero. It behaves as a single sample from this distribution.
– NCh
Nov 20 at 3:25












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There are several ways to look at it:




  • Let $f(x):=frac{pi^{-1}}{1+x^2}$ so $int_{-infty}^c xf(x) dx=-infty,,int_d^infty xf(x) dx$ for any $c,,dinmathbb{R}$. So if we choose $c<d$, you could argue the mean is $-infty+int_c^d xf(x) dx+infty$. In theory, you can get any value you like if you think the infinities cancel.

  • "But I'm integrating an odd function! That has to give me $0$!" Yes, if the two pieces you're cancelling are both finite. But $infty-infty$ is an indefinite form, so you can't use that theorem.

  • The characteristic function is $varphi(t):=exp -left|tright|$. If we average $n$ samples, the result has characteristic function $varphi^n(t/n)=varphi(t)$. It's immune to the CLT. Nor should you expect otherwise, without a finite and well-defined mean and variance. No $mu$, no $(X-mu^2)$, no variance. The characteristic function provides another way to look at it: we can't very well write $mu=varphi'(0)$ because the modulus has undefined derivative at $0$ (the one-sided limits $lim_{tto 0^pm}frac{|t|}{t}$ differ).

  • It can be shown, however, that the median of $n$ samples is asymptotically Normal for large $n$. (The proof is a bit more involved than a standard CLT argument for means; you can get an overview here.) By contrast, if you compute the mean of a gradually growing sample, it'll bounce around like crazy, because (as shown above) it's Cauchy-distributed.






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    1 Answer
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    There are several ways to look at it:




    • Let $f(x):=frac{pi^{-1}}{1+x^2}$ so $int_{-infty}^c xf(x) dx=-infty,,int_d^infty xf(x) dx$ for any $c,,dinmathbb{R}$. So if we choose $c<d$, you could argue the mean is $-infty+int_c^d xf(x) dx+infty$. In theory, you can get any value you like if you think the infinities cancel.

    • "But I'm integrating an odd function! That has to give me $0$!" Yes, if the two pieces you're cancelling are both finite. But $infty-infty$ is an indefinite form, so you can't use that theorem.

    • The characteristic function is $varphi(t):=exp -left|tright|$. If we average $n$ samples, the result has characteristic function $varphi^n(t/n)=varphi(t)$. It's immune to the CLT. Nor should you expect otherwise, without a finite and well-defined mean and variance. No $mu$, no $(X-mu^2)$, no variance. The characteristic function provides another way to look at it: we can't very well write $mu=varphi'(0)$ because the modulus has undefined derivative at $0$ (the one-sided limits $lim_{tto 0^pm}frac{|t|}{t}$ differ).

    • It can be shown, however, that the median of $n$ samples is asymptotically Normal for large $n$. (The proof is a bit more involved than a standard CLT argument for means; you can get an overview here.) By contrast, if you compute the mean of a gradually growing sample, it'll bounce around like crazy, because (as shown above) it's Cauchy-distributed.






    share|cite|improve this answer

























      up vote
      1
      down vote













      There are several ways to look at it:




      • Let $f(x):=frac{pi^{-1}}{1+x^2}$ so $int_{-infty}^c xf(x) dx=-infty,,int_d^infty xf(x) dx$ for any $c,,dinmathbb{R}$. So if we choose $c<d$, you could argue the mean is $-infty+int_c^d xf(x) dx+infty$. In theory, you can get any value you like if you think the infinities cancel.

      • "But I'm integrating an odd function! That has to give me $0$!" Yes, if the two pieces you're cancelling are both finite. But $infty-infty$ is an indefinite form, so you can't use that theorem.

      • The characteristic function is $varphi(t):=exp -left|tright|$. If we average $n$ samples, the result has characteristic function $varphi^n(t/n)=varphi(t)$. It's immune to the CLT. Nor should you expect otherwise, without a finite and well-defined mean and variance. No $mu$, no $(X-mu^2)$, no variance. The characteristic function provides another way to look at it: we can't very well write $mu=varphi'(0)$ because the modulus has undefined derivative at $0$ (the one-sided limits $lim_{tto 0^pm}frac{|t|}{t}$ differ).

      • It can be shown, however, that the median of $n$ samples is asymptotically Normal for large $n$. (The proof is a bit more involved than a standard CLT argument for means; you can get an overview here.) By contrast, if you compute the mean of a gradually growing sample, it'll bounce around like crazy, because (as shown above) it's Cauchy-distributed.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        There are several ways to look at it:




        • Let $f(x):=frac{pi^{-1}}{1+x^2}$ so $int_{-infty}^c xf(x) dx=-infty,,int_d^infty xf(x) dx$ for any $c,,dinmathbb{R}$. So if we choose $c<d$, you could argue the mean is $-infty+int_c^d xf(x) dx+infty$. In theory, you can get any value you like if you think the infinities cancel.

        • "But I'm integrating an odd function! That has to give me $0$!" Yes, if the two pieces you're cancelling are both finite. But $infty-infty$ is an indefinite form, so you can't use that theorem.

        • The characteristic function is $varphi(t):=exp -left|tright|$. If we average $n$ samples, the result has characteristic function $varphi^n(t/n)=varphi(t)$. It's immune to the CLT. Nor should you expect otherwise, without a finite and well-defined mean and variance. No $mu$, no $(X-mu^2)$, no variance. The characteristic function provides another way to look at it: we can't very well write $mu=varphi'(0)$ because the modulus has undefined derivative at $0$ (the one-sided limits $lim_{tto 0^pm}frac{|t|}{t}$ differ).

        • It can be shown, however, that the median of $n$ samples is asymptotically Normal for large $n$. (The proof is a bit more involved than a standard CLT argument for means; you can get an overview here.) By contrast, if you compute the mean of a gradually growing sample, it'll bounce around like crazy, because (as shown above) it's Cauchy-distributed.






        share|cite|improve this answer












        There are several ways to look at it:




        • Let $f(x):=frac{pi^{-1}}{1+x^2}$ so $int_{-infty}^c xf(x) dx=-infty,,int_d^infty xf(x) dx$ for any $c,,dinmathbb{R}$. So if we choose $c<d$, you could argue the mean is $-infty+int_c^d xf(x) dx+infty$. In theory, you can get any value you like if you think the infinities cancel.

        • "But I'm integrating an odd function! That has to give me $0$!" Yes, if the two pieces you're cancelling are both finite. But $infty-infty$ is an indefinite form, so you can't use that theorem.

        • The characteristic function is $varphi(t):=exp -left|tright|$. If we average $n$ samples, the result has characteristic function $varphi^n(t/n)=varphi(t)$. It's immune to the CLT. Nor should you expect otherwise, without a finite and well-defined mean and variance. No $mu$, no $(X-mu^2)$, no variance. The characteristic function provides another way to look at it: we can't very well write $mu=varphi'(0)$ because the modulus has undefined derivative at $0$ (the one-sided limits $lim_{tto 0^pm}frac{|t|}{t}$ differ).

        • It can be shown, however, that the median of $n$ samples is asymptotically Normal for large $n$. (The proof is a bit more involved than a standard CLT argument for means; you can get an overview here.) By contrast, if you compute the mean of a gradually growing sample, it'll bounce around like crazy, because (as shown above) it's Cauchy-distributed.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 at 10:38









        J.G.

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