What is a distribution of X given X − Y &gt; 0given two normally distributed variables X ∼ N ( μ X , σ X ) and Y ∼ N ( μ Y , σ Y ) that are independent ρ X Y = 0, what is a distribution of X given X &gt; Y.I have run several simulations in Matlab and X | X &gt; Y looks suspiciously normally distributed, but what are its mean and standard deviation?

Question

What is a distribution of X given   X  −  Y  &amp;gt;  0given two normally distributed variables   X  ∼      N    (      μ    X    ,      σ    X    ) and   Y  ∼      N    (      μ    Y    ,      σ    Y    ) that are independent       ρ          X      Y        =  0, what is a distribution of X given   X  &amp;gt;  Y.I have run several simulations in Matlab and   X      |    X  &amp;gt;  Y looks suspiciously normally distributed, but what are its mean and standard deviation?

Alvin Preston · Accepted Answer

Step 1Instead of thinking through the inequality   X  &amp;gt;  Y, see it as   Z  =  X  −  Y  &amp;gt;  0. Note that given that X and Y are both normal, Z is normal with mean       μ    x    −      μ    Y   and variance       σ    X    2    +      σ    Y    2   (assuming independence). Thus, you can calculate                    P        {        X        ≤        x                  |                X        &amp;gt;        Y        }                            =                              P            {            X            ≤            x            ,            X            &amp;gt;            Y            }                                P            {            X            &amp;gt;            Y            }                                                            =                              P            {            Y            &amp;lt;            X            ≤            x            }                                P            {            Z            &amp;gt;            0            }                              I assume you can work out   P  {  Z  &amp;gt;  0  } with ease, for it is the distribution function of a normal r.v.. The hard part is   P  {  Y  &amp;lt;  X  ≤  x  }. You can draw the area in             R        2   to see how to integrate it. I believe it is as follows (I changed x for z so there is no confusion in the integrands and limits):                    P        {        Y        &amp;lt;        X        ≤        z        }                            =                  ∫                      −            ∞                    z                          ∫                      −            ∞                    x                f        (        x        ,        y        )        d        y        d        x                                          =                  ∫                      −            ∞                    z                          ∫                      −            ∞                    x                f        (        x        )        f        (        y        )        d        y        d        x                (        Independence        )                                          =                  ∫                      −            ∞                    z                f        (        x        )                  ∫                      −            ∞                    x                f        (        y        )        d        y        d        x                                          =                  ∫                      −            ∞                    z                f        (        x        )                  F          Y                (        x        )        d        x            Step 2However, we could try to solve the simplest case, which is that where X and Y are standard normal r.v.. In such case, we have that                    P        {        Y        &amp;lt;        X        ≤        z        }                            =                  ∫                      −            ∞                    z                f        (        x        )                  F          Y                (        x        )        d        x                                          =                  ∫                      −            ∞                    z                ϕ        (        x        )        Φ        (        x        )        d        x                                          =                  ∫          0                      Φ            (            z            )                          u        d        u        =        0.5        Φ        (        z                  )          2                    where we used the variable change   u  =  Φ  (  x  ), and thus   d  u  =  ϕ  (  x  )  d  x, where   Φ is the distribution function of a standard normal and   ϕ its density. Moreover, now Z has mean 0 so   P  (  Z  &amp;gt;  0  )  =  0.5. Thus,  P  {  X  ≤  x      |    X  &amp;gt;  Y  }  =  0.5  Φ  (  x      )    2        /    0.5  =  Φ  (  x      )    2    .Then,   f  (  x      |    X  &amp;gt;  Y  )  =  2  Φ  (  x  )  ϕ  (  x  ). The image below shows a plot of the standard normal density   ϕ  (  x  ) and   f  (  x      |    X  &amp;gt;  Y  ) as given before. You can see they look very similar, as you guessed through simulation. However, the r.v.   X      |    X  &amp;gt;  Y is not normal even under the simplest of assumptions.

What is a distribution of X given X-Y>0

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