In probability theory and statistics, the Hardy distribution is a discrete probability distribution that expresses the probability of the hole score for a given golf player. It is based on G. H. Hardy's (Hardy, 1945) basic assumption that there are three types of shots:
good ( G ) {\displaystyle (G)} , bad ( B ) {\displaystyle (B)} and ordinary ( O ) {\displaystyle (O)} ,
where the probability of a good hit equals p {\displaystyle p} , the probability of a bad hit equals q {\displaystyle q} and the probability of an ordinary hit equals 1 − p − q {\displaystyle 1-p-q} . Hardy further assigned
a value of 2 to a good stroke, a value of 0 to a bad stroke and a value of 1 to a regular or ordinary stroke.
Once the sum of the values is greater than or equal to the value of the par of the hole, the number of strokes in question is equal to the score achieved on that hole. A birdie on a par three could then have come about in three ways: O G {\displaystyle OG} , G O {\displaystyle GO} and G G {\displaystyle GG} , respectively, with probabilities ( 1 − p − q ) p {\displaystyle (1-p-q)\,p} , p ( 1 − p − q ) {\displaystyle p\,(1-p-q)} and p 2 {\displaystyle p^{2}} .
Definitions
Probability mass function A discrete random variable X is said to have a Hardy distribution, with parameters p {\displaystyle p} , q {\displaystyle q} and m {\displaystyle m} if it has a probability mass function given by:
P ( X = n ) = ∑ j = m + 1 2 m ( n − 1 n − j ) q n − j ( A j , m + B j , m ) {\displaystyle P\left(X=n\right)\,=\,\sum _{j={\frac {m+1}{2}}}^{m}{n-1 \choose n-j}{q}^{n-j}\left(A_{j,m}+B_{j,m}\right)} if m is odd and
P ( X = n ) = ∑ j = m 2 m ( n − 1 n − j ) q n − j ( A j , m + B j , m ) {\displaystyle P\left(X=n\right)\,=\,\sum _{j={\frac {m}{2}}}^{m}{n-1 \choose n-j}{q}^{n-j}\left(A_{j,m}+B_{j,m}\right)} if m is even with
A j , m = ( j − 1 2 j − m − 1 ) p m − j + 1 ( 1 − p − q ) 2 j − m − 1 {\displaystyle A_{j,m}\,=\,{j-1 \choose 2\,j-m-1}{p}^{m-j+1}\left(1-p-q\right)^{2\,j-m-1}}
and
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