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Casino Games and the Central Limit Theorem

Date post: 04-Dec-2023
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UNLV Gaming Research & Review Journal Volume 17 Issue 2 Casino Games and the Central Limit Theorem 1 Ashok Singh, Ph. D.* $QWKRQ\ ) /XFDV 3K' Rohan J. Dalpatadu, Ph.D. Dennis J. Murphy, Ph.D. Abstract 7KH FHQWUDO OLPLW WKHRUHP LQ VLPSOH WHUPV VWDWHV WKDW WKH SUREDELOLW\ GLVWULEXWLRQ RI WKH PHDQ RI D UDQGRP VDPSOH IRU PRVW SUREDELOLW\ GLVWULEXWLRQV FDQ EH DSSUR[LPDWHG E\ D QRUPDO GLVWULEXWLRQ ZKHQ WKH QXPEHU RI REVHUYDWLRQV LQ WKH VDPSOH LV µVXI¿FLHQWO\¶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µFRQ¿GHQFH OLPLWV¶ IRU WRWDO FDVLQR ZLQ RU UHEDWHV RQ ORVVHV IRU WKHVH JDPHV :H ZLOO XVH 0RQWH &DUOR H[SHULPHQWV WR VLPXODWH WKH SOD\ RI D IHZ VORW JDPHV DQG WKH WDEOH JDPH RI EDFFDUDW WR HVWLPDWH WKH SUREDELOLW\ GLVWULEXWLRQ RI WKH PHDQ SD\RXW IRU VDPSOH VL]HV DV ODUJH DV DQG FRPSDUH LW WR WKH QRUPDO GLVWULEXWLRQ Introduction 7KH ODZ RI ODUJH QXPEHUV //1 DQG WKH FHQWUDO OLPLW WKHRUHP &/7 DUH WKH FRUQHUVWRQHV RI LQIHUHQWLDO VWDWLVWLFV 7KH //1 HQVXUHV WKDW DV WKH QXPEHU RI UDQGRP VDPSOHV FROOHFWHG IURP D SUREDELOLW\ GLVWULEXWLRQ LV LQFUHDVHG WKH VDPSOH PHDQ FRQYHUJHV WR WKH WUXH SRSXODWLRQ PHDQ DQG WKH &/7 JXDUDQWHHV WKDW WKH VDPSOLQJ GLVWULEXWLRQ RI WKH PHDQ ZLOO EH *DXVVLDQ SURYLGHG WKHUH DUH D VXI¿FLHQW QXPEHU RI LQGHSHQGHQW REVHUYDWLRQV 7KH ODZ RI ODUJH QXPEHUV ZDV XWLOL]HG E\ WKH &KHYDOLHU GH 0HUH DQ DVWXWH JDPEOHU VHH 0D[ZHOO DQG ¿UVW SURYHQ E\ -DNRE %HUQRXOOL LQ %DXHU ,Q $EUDKDP GH 0RLYUH LQWURGXFHG WKH FRQFHSW RI WKH *DXVVLDQ GLVWULEXWLRQ DV DQ DSSUR[LPDWLRQ WR WKH ELQRPLDO GLVWULEXWLRQ 7KLV UHVXOW QRZ FDOOHG WKH WKHRUHP RI GH 0RLYUH ± /DSODFH )HOOHU LV D VSHFLDO FDVH RI WKH &/7 7KH &/7 LV LQYRNHG LQ GHULYLQJ IRUPXODV IRU FRQ¿GHQFH LQWHUYDOV LQ HVWLPDWLRQ SUREOHPV DQG FULWLFDO UHJLRQV LQ K\SRWKHVLV WHVWLQJ SUREOHPV 7KHUH DUH PDQ\ SUDFWLFDO DSSOLFDWLRQV RI WKH FHQWUDO OLPLW WKHRUHP 7KH &/7 SURYLGHV MXVWL¿FDWLRQ IRU PDQ\ SURFHGXUHV LQ VWDWLVWLFDO SURFHVV FRQWURO DQG VWDWLVWLFDO TXDOLW\ 7KLV UHVHDUFK ZDV FRQGXFWHG ZLWK IXQGLQJ VXSSRUW IURP WKH &DHVDUV )RXQGDWLRQ $VKRN 6LQJK 3K ' 3URIHVVRU 8QLYHUVLW\ RI 1HYDGD /DV 9HJDV :LOOLDP ) +DUUDK &ROOHJH RI +RWHO Administration 0DU\ODQG 3DUNZD\ /DV9HJDV 19 Email: DNVLQJK#XQOYQHYDGDHGX $QWKRQ\ ) /XFDV 3K' 3URIHVVRU :LOOLDP ) +DUUDK &ROOHJH RI +RWHO Administration 8QLYHUVLW\ RI 1HYDGD /DV 9HJDV (PDLO $)/#FR[QHW 5RKDQ - 'DOSDWDGX 3K' $VVRFLDWH 3URIHVVRU 8QLYHUVLW\ RI 1HYDGD /DV 9HJDV Department of Mathematical Sciences 0DU\ODQG 3DUNZD\ /DV9HJDV 19 Email: GDOSDWDG#XQOYQHYDGDHGX 'HQQLV - 0XUSK\ 3K' Senior Statistician $',19 ,QF Email: GMPXUSK\#\DKRRFRP &RUUHVSRQGLQJ DXWKRU &DVLQR *DPHV DQG WKH &HQWUDO /LPLW 7KHRUHP
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��UNLV Gaming Research & Review Journal � Volume 17 Issue 2

Casino Games and the Central Limit Theorem1

Ashok Singh, Ph. D.* $QWKRQ\�)��/XFDV��3K�'�

Rohan J. Dalpatadu, Ph.D.Dennis J. Murphy, Ph.D.

Abstract

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Introduction

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50 UNLV Gaming Research & Review Journal � Volume 17 Issue 2

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