Let X and Y be random variables. The Ky Fan metric is defined as: d ( X , Y ) := min { ϵ &gt; 0 : P ( | X − Y | &gt; ϵ ) ≤ ϵ }I want to show it is indeed a metric, for which I need to show that it satisfies triangle inequality.Set d ( X , Y ) = ϵ 1 , d ( Y , Z ) = ϵ 2 ,and d ( Z , X ) = ϵ 3 .I approached this by trying to show that ( ϵ 1 + ϵ 2 ) ∈ { ϵ &gt; 0 : P ( | X − Z | &gt; ϵ ) ≤ ϵ }. But to show that, I had to show P ( | X − Y | &gt; ϵ 1 ) + P ( | Y − Z | &gt; ϵ 2 ) ≥ P ( | X − Z | &gt; ϵ 1 + ϵ 2 ), which I could not. Can you help in showing this?

Question

Let   X and   Y be random variables. The Ky Fan metric is defined as:  d  (  X  ,  Y  )  :=  min  {  ϵ  &amp;gt;  0  :  P  (      |    X  −  Y      |    &amp;gt;  ϵ  )  ≤  ϵ  }I want to show it is indeed a metric, for which I need to show that it satisfies triangle inequality.Set   d  (  X  ,  Y  )  =      ϵ    1    ,  d  (  Y  ,  Z  )  =      ϵ    2  ,and   d  (  Z  ,  X  )  =      ϵ    3   .I approached this by trying to show that   (      ϵ    1    +      ϵ    2    )  ∈  {  ϵ  &amp;gt;  0  :  P  (      |    X  −  Z      |    &amp;gt;  ϵ  )  ≤  ϵ  }. But to show that, I had to show   P  (      |    X  −  Y      |    &amp;gt;      ϵ    1    )  +  P  (      |    Y  −  Z      |    &amp;gt;      ϵ    2    )  ≥  P  (      |    X  −  Z      |    &amp;gt;      ϵ    1    +      ϵ    2    ), which I could not. Can you help in showing this?

arhaitategr · Accepted Answer

If       |    X  −  Y      |    ≤      ϵ    1   and       |    Y  −  Z      |    ≤      ϵ    2  , then we have       |    X  −  Z      |    ≤      ϵ    1    +      ϵ    2  .Hence if       |    X  −  Z      |    &amp;gt;      ϵ    1    +      ϵ    2  , then       |    X  −  Y      |    &amp;gt;      ϵ    1   or       |    Y  −  Z      |    &amp;gt;      ϵ    2  .  {  ω  :      |    X  (  ω  )  −  Z  (  ω  )      |    &amp;gt;      ϵ    1    +      ϵ    2    }  ⊆  {  ω  :      |    X  (  ω  )  −  Y  (  ω  )      |    &amp;gt;      ϵ    1    }  ∪  {  ω  :      |    Y  (  ω  )  −  Z  (  ω  )      |    &amp;gt;      ϵ    2    }Hence, by the union bound,  P  (      |    X  −  Y      |    &amp;gt;      ϵ    1    )  +  P  (      |    Y  −  Z      |    &amp;gt;      ϵ    2    )  ≥  P  (      |    X  −  Z      |    &amp;gt;      ϵ    1    +      ϵ    2    )

Let X and Y be random variables. The Ky Fan metric is defined as: d ( X ,

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