Geometric Distribution - transformation Y = x / ( x + 1 )I honestly have no idea where to start with this problem:Suppose X has the geometric distribution with PMF f ( x ) = 1 3 ( 2 3 ) x , x = 1 , 2 , . . . . .Determine the probability distribution of Y = x / ( x + 1 ) . Note that both X and Y are discrete random variables. To specify the probability distribution of Y, specify its PMF.

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Geometric Distribution - transformation   Y  =  x      /    (  x  +  1  )I honestly have no idea where to start with this problem:Suppose X has the geometric distribution with PMF  f  (  x  )  =      1    3              (              2        3            )        x    ,  x  =  1  ,  2  ,  .  .  .  .  .Determine the probability distribution of   Y  =  x      /    (  x  +  1  )  . Note that both X and Y are discrete random variables. To specify the probability distribution of Y, specify its PMF.

Hassan Watkins · Accepted Answer

Step 1To specify a variable&#039;s PMF, you need two things: (a) a list of what values it can be, and a corresponding list of probabilities of it being those values.You already have the PMF for X, but let&#039;s make a table out of it to make it a bit more concrete.                    x                    p        (        x        )                            0                    1                  /                3                            1                              1          3                ⋅                  2          3                =                  2          9                                    2                              1          3                ⋅        (                  2          3                          )          2                =                  4          27                                    ⋮                    ⋮            I obtained the probabilities, denote above as p(x), by plugging each value of x into the given PMF.Step 2Once you have these, you now have a complete listing of what Y can be, along with all its probabilities. For instance, if   X  =  0, then   Y  =      X          X      +      1        =      0    1    =  0 as well; this will occur with probability 1/3. If   X  =  1, then   Y  =      1    2  , which will occur with probability 2/9, and so forth. Using this idea, you could make a table that looks just like the above, to get the the PMF for Y.Of course, it may be more desirable to put this into a formula; you may want something compact like the expression you gave for f(x). I claim that the clearest way to see what such a formula should be is to write out the first 3 or 4 lines of a table like the above, though.

Geometric Distribution - transformation Y=x/(x+1)

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