Interpreting probability question in terms of measure theoryLet X and Y be real valued random variables with joint pdf f X , Y ( x , y ) = { 1 4 ( x + y ) , 0 ≤ x ≤ y ≤ 2 0 , otherwise Calculate the probability P { Y &lt; 2 X }I am trying to view this problem in a measure-theoretic perspective and I am wondering if I am thinking about this properly.Let ( Ω , F , P ) be a probability space on which we define two random variables (measurable functions) X and Y. We then consider their joint distribution, the pushforward measure of the random variable T ( ω ) = ( X ( ω ) , Y ( ω ) ) on ( R 2 , B × B ) defined by T ⋆ P ( A ) = P ( T − 1 ( A ) ) = P ( ( X ( ω ) , Y ( ω ) ) ∈ A ) .Then by the question, we have that d T ⋆ P d λ = f X , Y i.e. the Radon-Nikodym derivative of the pushforward of T with respect to the Lebesgue measure is the pdf of (X,Y).How can I then calculate P { Y &lt; 2 X }? Somehow I have to relate the probability of this set to the pushforward measuere of which I know the density. What is the theorem that allows me to relate these two measures?Essentially, im looking for the measure of the set { ω : Y ( ω ) &lt; 2 X ( ω ) } ⊂ Ω. I know the density of a measure on R 2 which is a different set than Ω. How do I know that { ( x , y ) : y &lt; 2 x } ⊂ R 2 is the subset of R2 with which I need to integrate over?

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Interpreting probability question in terms of measure theoryLet X and Y be real valued random variables with joint pdf      f          X      ,      Y        (  x  ,  y  )  =      {                                        1            4                    (          x          +          y          )          ,                          0          ≤          x          ≤          y          ≤          2                                      0          ,                           otherwise                         Calculate the probability       P    {  Y  &amp;lt;  2  X  }I am trying to view this problem in a measure-theoretic perspective and I am wondering if I am thinking about this properly.Let   (  Ω  ,      F    ,      P    ) be a probability space on which we define two random variables (measurable functions) X and Y. We then consider their joint distribution, the pushforward measure of the random variable   T  (  ω  )  =  (  X  (  ω  )  ,  Y  (  ω  )  ) on       (                  R            2        ,          B        ×          B        )   defined by      T    ⋆        P    (  A  )  =      P        (          T              −        1              (    A    )    )    =      P    (  (  X  (  ω  )  ,  Y  (  ω  )  )  ∈  A  )  .Then by the question, we have that                     d                  T          ⋆                          P                            d        λ              =      f          X      ,      Y       i.e. the Radon-Nikodym derivative of the pushforward of T with respect to the Lebesgue measure is the pdf of (X,Y).How can I then calculate       P    {  Y  &amp;lt;  2  X  }? Somehow I have to relate the probability of this set to the pushforward measuere of which I know the density. What is the theorem that allows me to relate these two measures?Essentially, im looking for the measure of the set   {  ω  :  Y  (  ω  )  &amp;lt;  2  X  (  ω  )  }  ⊂  Ω. I know the density of a measure on             R        2   which is a different set than   Ω. How do I know that   {  (  x  ,  y  )  :  y  &amp;lt;  2  x  }  ⊂            R        2   is the subset of R2 with which I need to integrate over?

Savion Chaney · Accepted Answer

Step 1How do I know that   {  (  x  ,  y  )  :  y  &amp;lt;  2  x  }  ⊂            R        2   is the subset of             R        2   with which I need to integrate over?This follows from the definition of the term &quot;density&quot; under the measure-theoretic framework. See, for example, Probability and Measure (Section 20, equation (20.9)), which I quote as follows:A random variable and its distribution have density f with respect to Lebesgue measure if f is a nonnegative Borel function on             R        1   and  P  [  X  ∈  A  ]  =  μ  (  A  )  =      ∫    A    f  (  x  )  d  x  ,    A  ∈            R        1    .Although this (fundamental) definition applies to a single random variable, it clearly can be generalized to random vectors, as the author later sketched in the same section:The distribution (of a random vector) may as for the line be discrete in the sense of having countable support. It may have density f with respect to k-dimensional Lebesgue measure:   μ  (  A  )  =      ∫    A    f  (  x  )  d  x. As in the case   k  =  1, the distribution μ is more fundamental than the distribution function F, and usually   μ is described not by F but by a density or by discrete probabilities.In your case, your A is clearly the two-dimensional Borel set   {  (  x  ,  y  )  :  y  &amp;lt;  2  x  }, whence, according to the definition above,                    P        [        (        X        ,        Y        )        ∈        A        ]        =        μ        (        A        )        =                  ∫          A                f        (        x        ,        y        )        d        x        d        y        .

Kayden Mills · Accepted Answer

Step 1In probability space   (  Ω  ,      F    ,      P    ) with variables X,Y with joint distribution measure       μ          X      ,      Y      :                    P        {        Y        &amp;lt;        2        X        }                            =                  ∫          Ω                                      I                                y            &amp;lt;            2            x                                  d                  P                                                  =                  ∫                                                    R                            2                                                            I                                y            &amp;lt;            2            x                                  d                  μ                      X            ,            Y                              Step 2The definition of a joint density function is the nonnegative Borel measurable       f          X      ,      Y        (  x  ,  y  ) such that:                              μ                      X            ,            Y                          (        C        )        =                  ∫                      −            ∞                    ∞                          ∫                      −            ∞                    ∞                                      I                    C                (        x        ,        y        )                  f                      X            ,            Y                          (        x        ,        y        )                d        y                d        x                C        ∈                  B                (                              R                    2                )            Step 3Then   d      μ          X      ,      Y        =      f          X      ,      Y          d  x    d  y, so plugging that in:                    P        {        Y        &amp;lt;        2        X        }                            =                  ∫                                                    R                            2                                                            I                                y            &amp;lt;            2            x                                            f                      X            ,            Y                                  d        x                d        y                                          =                  ∫                                                    R                            2                                                            I                                y            &amp;lt;            2            x                                                        I                                0            ≤            x            ≤            y            ≤            2                                    1          4                (        x        +        y        )                d        x                d        y                                          =                  ∫          0          2                          ∫                      y                          /                        2                    y                          1          4                (        x        +        y        )                d        x                d        y                                          =                  ∫          0          2                          7          32                          y          2                        d        y                                          =                  7          12

Let X and Y be real valued random variables with joint pdf. f_{X,Y} (x,y)={(1/4(x+y),0<=x<=y<=2,,,),(0,otherwise,,,):}. Calculate the probability P{Y<2X}.

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