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Find the conditional probability density function of $Y$ given $X=x$


Probability Density Function - GammaFinding quantiles based on probability density functionsProbability density function of random variable $X-Y$Find the conditional probability of a probability density functionConditional Distribution from Conditional densityFind the marginal and conditional densities without explicitly having the joint density?Find the conditional probability density function of $X$ given $Y=y$Computing probability function of $Y$ in terms of $f_X$How can I get the probability density function?Compute the probability density function of Y













0












$begingroup$


Suppose $Y$ is a continuous random variable with probability density function $f(y)=192over y^4$, for $ygeq 4$ ($0$ otherwise). If the conditional distribution of X given $Y = y$ is a uniform distribution on $[0, y]$. Find the conditional probability density function of $Y$ given $X = x$.



My clue is that:
$$f_Y(x|y)=f_X,Y(x,y)overf_Y(y)=1over y$$
then we get $$f_X,Y(x,y)=192over y^5$$
$$f_Y(y|x)=f_X,Y(x,y)overf_X(x)$$
But I don't know how to find the domain of $Y$ in terms of $X$ to get $f_X(x)$.

Am I on the right track?










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    Suppose $Y$ is a continuous random variable with probability density function $f(y)=192over y^4$, for $ygeq 4$ ($0$ otherwise). If the conditional distribution of X given $Y = y$ is a uniform distribution on $[0, y]$. Find the conditional probability density function of $Y$ given $X = x$.



    My clue is that:
    $$f_Y(x|y)=f_X,Y(x,y)overf_Y(y)=1over y$$
    then we get $$f_X,Y(x,y)=192over y^5$$
    $$f_Y(y|x)=f_X,Y(x,y)overf_X(x)$$
    But I don't know how to find the domain of $Y$ in terms of $X$ to get $f_X(x)$.

    Am I on the right track?










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      Suppose $Y$ is a continuous random variable with probability density function $f(y)=192over y^4$, for $ygeq 4$ ($0$ otherwise). If the conditional distribution of X given $Y = y$ is a uniform distribution on $[0, y]$. Find the conditional probability density function of $Y$ given $X = x$.



      My clue is that:
      $$f_Y(x|y)=f_X,Y(x,y)overf_Y(y)=1over y$$
      then we get $$f_X,Y(x,y)=192over y^5$$
      $$f_Y(y|x)=f_X,Y(x,y)overf_X(x)$$
      But I don't know how to find the domain of $Y$ in terms of $X$ to get $f_X(x)$.

      Am I on the right track?










      share|cite|improve this question











      $endgroup$




      Suppose $Y$ is a continuous random variable with probability density function $f(y)=192over y^4$, for $ygeq 4$ ($0$ otherwise). If the conditional distribution of X given $Y = y$ is a uniform distribution on $[0, y]$. Find the conditional probability density function of $Y$ given $X = x$.



      My clue is that:
      $$f_Y(x|y)=f_X,Y(x,y)overf_Y(y)=1over y$$
      then we get $$f_X,Y(x,y)=192over y^5$$
      $$f_Y(y|x)=f_X,Y(x,y)overf_X(x)$$
      But I don't know how to find the domain of $Y$ in terms of $X$ to get $f_X(x)$.

      Am I on the right track?







      probability






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 14 at 6:15









      Rócherz

      2,9863821




      2,9863821










      asked Oct 1 '18 at 18:06









      Yibei HeYibei He

      3139




      3139




















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Note that the expression for the joint density $f_X,Y(x,y)$ that you computed only holds for $x$ and $y$ satisfying $x ge 0$ and $y ge maxx, 4$; it is zero otherwise. Thus for $x ge 0$ we have
          $$f_X(x) = int_-infty^infty f_X,Y(x,z) , dz = int_maxx, 4^infty frac192z^5 , dz.$$




          Continuing the computation leads to
          $$f_X(x) = 48 maxx, 4^-4 = begincases48 / 4^4 & x in [0, 4] \ 48 / x^4 & x > 4endcases$$
          So,
          $$int_0^infty f_X(x) , dx = 4 cdot frac484^4 + int_4^infty frac48x^4 , dx
          = frac34 + frac164^3 = 1.$$






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
            $endgroup$
            – Yibei He
            Oct 1 '18 at 20:13










          • $begingroup$
            @YibeiHe Good catch; I've edited my answer.
            $endgroup$
            – angryavian
            Oct 1 '18 at 22:12










          • $begingroup$
            I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
            $endgroup$
            – Yibei He
            Oct 2 '18 at 2:44










          • $begingroup$
            @YibeiHe See my edit.
            $endgroup$
            – angryavian
            Oct 2 '18 at 2:51










          Your Answer





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          1 Answer
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          active

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          1












          $begingroup$

          Note that the expression for the joint density $f_X,Y(x,y)$ that you computed only holds for $x$ and $y$ satisfying $x ge 0$ and $y ge maxx, 4$; it is zero otherwise. Thus for $x ge 0$ we have
          $$f_X(x) = int_-infty^infty f_X,Y(x,z) , dz = int_maxx, 4^infty frac192z^5 , dz.$$




          Continuing the computation leads to
          $$f_X(x) = 48 maxx, 4^-4 = begincases48 / 4^4 & x in [0, 4] \ 48 / x^4 & x > 4endcases$$
          So,
          $$int_0^infty f_X(x) , dx = 4 cdot frac484^4 + int_4^infty frac48x^4 , dx
          = frac34 + frac164^3 = 1.$$






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
            $endgroup$
            – Yibei He
            Oct 1 '18 at 20:13










          • $begingroup$
            @YibeiHe Good catch; I've edited my answer.
            $endgroup$
            – angryavian
            Oct 1 '18 at 22:12










          • $begingroup$
            I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
            $endgroup$
            – Yibei He
            Oct 2 '18 at 2:44










          • $begingroup$
            @YibeiHe See my edit.
            $endgroup$
            – angryavian
            Oct 2 '18 at 2:51















          1












          $begingroup$

          Note that the expression for the joint density $f_X,Y(x,y)$ that you computed only holds for $x$ and $y$ satisfying $x ge 0$ and $y ge maxx, 4$; it is zero otherwise. Thus for $x ge 0$ we have
          $$f_X(x) = int_-infty^infty f_X,Y(x,z) , dz = int_maxx, 4^infty frac192z^5 , dz.$$




          Continuing the computation leads to
          $$f_X(x) = 48 maxx, 4^-4 = begincases48 / 4^4 & x in [0, 4] \ 48 / x^4 & x > 4endcases$$
          So,
          $$int_0^infty f_X(x) , dx = 4 cdot frac484^4 + int_4^infty frac48x^4 , dx
          = frac34 + frac164^3 = 1.$$






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
            $endgroup$
            – Yibei He
            Oct 1 '18 at 20:13










          • $begingroup$
            @YibeiHe Good catch; I've edited my answer.
            $endgroup$
            – angryavian
            Oct 1 '18 at 22:12










          • $begingroup$
            I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
            $endgroup$
            – Yibei He
            Oct 2 '18 at 2:44










          • $begingroup$
            @YibeiHe See my edit.
            $endgroup$
            – angryavian
            Oct 2 '18 at 2:51













          1












          1








          1





          $begingroup$

          Note that the expression for the joint density $f_X,Y(x,y)$ that you computed only holds for $x$ and $y$ satisfying $x ge 0$ and $y ge maxx, 4$; it is zero otherwise. Thus for $x ge 0$ we have
          $$f_X(x) = int_-infty^infty f_X,Y(x,z) , dz = int_maxx, 4^infty frac192z^5 , dz.$$




          Continuing the computation leads to
          $$f_X(x) = 48 maxx, 4^-4 = begincases48 / 4^4 & x in [0, 4] \ 48 / x^4 & x > 4endcases$$
          So,
          $$int_0^infty f_X(x) , dx = 4 cdot frac484^4 + int_4^infty frac48x^4 , dx
          = frac34 + frac164^3 = 1.$$






          share|cite|improve this answer











          $endgroup$



          Note that the expression for the joint density $f_X,Y(x,y)$ that you computed only holds for $x$ and $y$ satisfying $x ge 0$ and $y ge maxx, 4$; it is zero otherwise. Thus for $x ge 0$ we have
          $$f_X(x) = int_-infty^infty f_X,Y(x,z) , dz = int_maxx, 4^infty frac192z^5 , dz.$$




          Continuing the computation leads to
          $$f_X(x) = 48 maxx, 4^-4 = begincases48 / 4^4 & x in [0, 4] \ 48 / x^4 & x > 4endcases$$
          So,
          $$int_0^infty f_X(x) , dx = 4 cdot frac484^4 + int_4^infty frac48x^4 , dx
          = frac34 + frac164^3 = 1.$$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Oct 2 '18 at 2:51

























          answered Oct 1 '18 at 18:10









          angryavianangryavian

          42.2k23481




          42.2k23481











          • $begingroup$
            So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
            $endgroup$
            – Yibei He
            Oct 1 '18 at 20:13










          • $begingroup$
            @YibeiHe Good catch; I've edited my answer.
            $endgroup$
            – angryavian
            Oct 1 '18 at 22:12










          • $begingroup$
            I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
            $endgroup$
            – Yibei He
            Oct 2 '18 at 2:44










          • $begingroup$
            @YibeiHe See my edit.
            $endgroup$
            – angryavian
            Oct 2 '18 at 2:51
















          • $begingroup$
            So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
            $endgroup$
            – Yibei He
            Oct 1 '18 at 20:13










          • $begingroup$
            @YibeiHe Good catch; I've edited my answer.
            $endgroup$
            – angryavian
            Oct 1 '18 at 22:12










          • $begingroup$
            I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
            $endgroup$
            – Yibei He
            Oct 2 '18 at 2:44










          • $begingroup$
            @YibeiHe See my edit.
            $endgroup$
            – angryavian
            Oct 2 '18 at 2:51















          $begingroup$
          So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
          $endgroup$
          – Yibei He
          Oct 1 '18 at 20:13




          $begingroup$
          So when you integral the joint pdf from x to ∞, you assumed that x is greater than 4, right? Otherwise it will be from 4 to ∞. But does it contradict with 0≤x?
          $endgroup$
          – Yibei He
          Oct 1 '18 at 20:13












          $begingroup$
          @YibeiHe Good catch; I've edited my answer.
          $endgroup$
          – angryavian
          Oct 1 '18 at 22:12




          $begingroup$
          @YibeiHe Good catch; I've edited my answer.
          $endgroup$
          – angryavian
          Oct 1 '18 at 22:12












          $begingroup$
          I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
          $endgroup$
          – Yibei He
          Oct 2 '18 at 2:44




          $begingroup$
          I'm still quite confused about the domain. Do you mean $f_X(x)$ will have two case, one is x=[0,4] and the other is x=[4,∞]? If I get $f_X(x)$ in this way, when I integral $f_X(x)$ in both case, I can't get the summation to 1. How does that happen?
          $endgroup$
          – Yibei He
          Oct 2 '18 at 2:44












          $begingroup$
          @YibeiHe See my edit.
          $endgroup$
          – angryavian
          Oct 2 '18 at 2:51




          $begingroup$
          @YibeiHe See my edit.
          $endgroup$
          – angryavian
          Oct 2 '18 at 2:51

















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