A -Statistical extension of the Korovkin type approximation by Erkus E., Duman O.

By Erkus E., Duman O.

During this paper, utilizing the idea that ofA-statistical convergence that is a regular(non-matrix) summability strategy, we receive a normal Korovkin sort approximation theorem which matters the matter of approximating a functionality f through a chain {Lnf } of optimistic linear operators.

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1), the sample space S1 consists of eight S1 = {HHH, ... , TTT}. v. giving the number of heads obtained, find (a) In the experiment of tossing a fair coin three times (Prob. equally likely sample points P(X=2); (b) P(X< 2). (a) Let AC S1 be the event defined by X= 2. Then, from Prob. 1, we have A = (X= 2) = {l;: X(l;) = 2} = {HHT, HTH, THH} Since the sample points are equally likely, we have 3 P(X=2)=P(A)=8 (b) Let BC S1 be the event defined by X< 2. 3 A. v. X is determined by the behavior of Fx(x).

5), S nB=B. 55), P[(A1 UA2)nB] P(AI UA2 IB)= P(B) Now by Eqs. 17), we have and AI nAz = 0 impliesthat (AI n B) n (Az nB) = 0. 44. Find P(A I B) if (a) An B = 0, (b) A c B, and (c) B c A. (a) IfAn B= 0,thenP(An B) = P(0) = 0. 45. Show that if P(A I B)> P(A), then P(B I A)> P(B). �IB )=P(AnB) >P(A), thenP(An' B)>P(A)P(B). 46. B) By Eqs. 7), we have AUA =S, Then by Eqs. 98), we get P(AUAI B)= P(SiB)=1= P(AiB) + P(AiB) Thus we obtain P(AiB)=1- P(AiB) note that Eq. 99) is similar to property 1 (Eq. 39)).

688 16 16 16 16 16 16 0 16 16 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Fig. 41. Consider the experiment of tossing a fair coin repeatedly and counting the number of tosses required until the first head appears. (a) Find the sample space of the experiment. (b) Find the probability that the first head appears on the (c) Verify that P(S) = 1. kth toss. CHAPTER 1 Probability (a) The sample space of this experiment is where (b) ek is the elementary event that the first head appears on the kth toss.

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