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Combinatorial argument probabilities
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[QUOTE="Dassinia, post: 5003367, member: 487299"] [B]Hello 1. Homework Statement [/B] Consider f[SUB]k[/SUB] N*→N, k≥0 f[SUB]k[/SUB](n)=Σ j[SUP]k[/SUP] n>0 We're looking to establish these identities, using a combinatorial argument f[SUB]k[/SUB](n)=n if k=0 f[SUB]k[/SUB](n)= 1/(k+1) [ n[SUP]k+1[/SUP] + Σ ( i , k+1) (-1)[SUP]k+1-i[/SUP]f[SUB]i[/SUB](n) ] if k>0 the sum is about i from i=0 to i=k-1 and (i, k+1) is the combination. Consider the experience of k+1 drawing with handoff from a box of n balls numbered Ej : the maximum of the values obtained on the k+1 draw is j , j=1,...,n Fj : The drawing are made on {1,...,j} j=1,...,n Fo=[ATTACH=full]201824[/ATTACH] 1. Find |Ω| 2. Write Ω as an operation on sets E1,...,En 3. Write Ej as an operation on sets F[SUB]j-1[/SUB], F[SUB]j[/SUB] j=1...n 4. Find |Fj| j=0...n 5. Find |Ej| j=1...n 6. Find |Ω| through the calculation of |Ej| j=1...n and using the binomial theorem [h2]Homework Equations[/h2] (x+y)[SUP]n[/SUP]=∑(k, n)x[SUP]n-k[/SUP]y[SUP]k[/SUP] [h2]The Attempt at a Solution[/h2] 1. |Ω|=n[SUP]k+1[/SUP] 2. Ω=∩ Ej for j=1,...,n 3. I just don't get what Fj represents..? :oldconfused: Thanks [/QUOTE]
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