Prove that for every two independent identically distributed real random variables X and Y, P r [ | X − Y | ≤ 2 ] ≤ 3 P r [ | X − Y | ≤ 1 ] .

Question

Prove that for every two independent identically distributed real random variables   X and   Y,  P  r  [      |    X  −  Y      |    ≤  2  ]  ≤  3  P  r  [      |    X  −  Y      |    ≤  1  ]  .

moralizoL31 · Accepted Answer

Prove it for the case when   Z  =      |    X  −  Y      |   takes only integer values.Let       q    i    =  P  (  Z  =  i  ) for   i  =  0  ,  1  ,  …. Then, we need to show that                     q        0            +              q        1                            q        0            +              q        1            +              q        2              ≥      1    3  . This follows from the observation that   2      q    0    ≥      q    i   for all   i. This follows from Cauchy Schwarz inequality. Then,                    3        (                  q          0                +                  q          1                )                            ≥        (                  q          0                +                  q          1                +                  q          2                )                            2        (                  q          0                +                  q          1                )                            ≥                  q          2                    which is true since   2      q    0    ≥      q    2  . In the case of   Z being real, I tried mimicking the proof above but the details don&#039;t quite work out. In this case, Cauchy-Schwarz still implies that       f    Z    (  z  )  ≤  2      f    Z    (  0  ) for all   z. However, the proof seems to need one more estimation along the lines of       ∫    0    1        f    Z    (  z  )  d  z  ≥      f    Z    (  0  ).

funnyantyLEy · Accepted Answer

You seem to have the answer (for integer distributions) - need to show  3  (      q    0    +      q    1    )  ≥      q    0    +      q    1    +      q    2  , which is true since   2      q    0    ≥      q    i   for any   i.

Prove that for every two independent identically distributed real random variables X and Y, Pr[|X−Y|≤2]≤3Pr[|X−Y|≤1].

Answered question

Answer & Explanation

New Questions in Algebra I