Let X 1 , … , X n be independent random variables with densities: f X k ( x ∣ θ ) = { e k θ − x x ≥ k θ 0 otherwise Find the pmf for T = min k ( X k k ) .

Question

Let       X    1    ,  …  ,      X    n   be independent random variables with densities:      f                  X        k              (  x  ∣  θ  )  =      {                                        e                          k              θ              −              x                                                x          ≥          k          θ                                      0                          otherwise                        Find the pmf for   T  =      min          k            (                  X        k            k        )  .

erlent00s · Accepted Answer

Notice that  Pr  (      Y    k    &amp;gt;  x  )  =      ∫    x    ∞    k      e          k      (      θ      −      w      )          d  w  =      e          k      (      θ      −      x      )        .So                    Pr        (        min        &amp;gt;        x        )                            =        Pr        (                  for all                       k                    ,                           Y          k                &amp;gt;        x        )                                          =                  e                      θ            −            x                                    e                      2            (            θ            −            x            )                                    e                      3            (            θ            −            x            )                          ⋯                  e                      n            (            θ            −            x            )                                                            =                  e                      n            (            n            +            1            )            (            θ            −            x            )                          /                        2                          .            Consequently for   x  &amp;gt;  θ, we have      f    min    (  x  )  =            d              d        x              (  1  −      e          n      (      n      +      1      )      (      θ      −      x      )              /            2        )  =            n      (      n      +      1      )        2    ⋅      e          n      (      n      +      1      )      (      θ      −      x      )              /            2        .Dividing by       e          θ      −      x            e          2      θ      −      x            e          3      θ      −      x        ⋯      e          n      θ      −      x      , one sees that the   θ cancels and the   x doesn&#039;t.

Let X_1,…,X_n be independent random variables with densities: f_X_k(x∣theta)={ e^(k theta−x) x>=k theta, 0 otherwise Find the pmf for T=min_ k (X_k/k).

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