Hardy distribution information

Hardy Distribution
	Probability mass function The horizontal axis represents the hole score n. The vertical axis represents the probability of the hole score n given the par of the hole and the probabilities p = 0.20 and q = 0.10. The blue points represent the probabilities for a par three, the green points for a par four and the red points for a par five The function is defined only at integer values of n. The connecting lines are only guides for the eye.
	Cumulative distribution function The horizontal axis represents the hole score n. The vertical axis represents the cumulative probability of the hole score n given the par of the hole and the probabilities p = 0.20 and q = 0.10. The blue points represent the probabilities for a par three, the green points for a par four and the red points for a par five. The cumulative probability density (CDF) is discontinuous at the integers of n and flat everywhere else because a variable that is Hardy distributed takes on only integer values.
Notation
Parameters	, and
Support	(Natural numbers starting from 1)
PMF	For m is odd: For m is even: with and
Mean
MGF	For m is odd: For m is even: with and

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 Hardy's (Hardy, 1945) basic assumption that there are three types of shots:

good  $(G)$ , 
bad  $(B)$  and 
ordinary  $(O)$ ,

where the probability of a good hit equals $p$ , the probability of a bad hit equals $q$ and the probability of an ordinary hit equals $1-p-q$ . 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: $OG$ , $GO$ and $GG$ , respectively, with probabilities $(1-p-q)\,p$ , $p\,(1-p-q)$ and $p^{2}$ .

Probability mass function The horizontal axis represents the hole score $n$ . The vertical axis represents the probability of the hole score $n$ given the par of the hole and the probabilities $p$ = 0.20 and $q$ = 0.10. The blue points represent the probabilities for a par three, the green points for a par four and the red points for a par five The function is defined only at integer values of $n$ . The connecting lines are only guides for the eye.
Cumulative distribution function The horizontal axis represents the hole score $n$ . The vertical axis represents the cumulative probability of the hole score $n$ given the par of the hole and the probabilities $p$ = 0.20 and $q$ = 0.10. The blue points represent the probabilities for a par three, the green points for a par four and the red points for a par five. The cumulative probability density (CDF) is discontinuous at the integers of $n$ and flat everywhere else because a variable that is Hardy distributed takes on only integer values.
Notation	$\operatorname {Hardy} (p,q;m)$
Parameters	$p,q\in (0,1)$ , $p+q\in (0,1)$ and $m=1,2,3,\dots$
Support	$n\in \mathbb {N} _{0}$ (Natural numbers starting from 1)
PMF	For m is odd: $P\left(X=n\right)\,=\,\sum _{j={\frac {m+1}{2}}}^{m}{n-1 \choose n-j}{q}^{n-j}\left(A_{j,m}+B_{j,m}\right)$ For m is even: $P\left(X=n\right)\,=\,\sum _{j={\frac {m}{2}}}^{m}{n-1 \choose n-j}{q}^{n-j}\left(A_{j,m}+B_{j,m}\right)$ with $A_{j,m}\,=\,{j-1 \choose 2\,j-m-1}{p}^{m-j+1}\left(1-p-q\right)^{2\,j-m-1}$ and $B_{j,m}\,=\,{j \choose 2\,j-m}{p}^{m-j}\left(1-p-q\right)^{2\,j-m}$
Mean	$\,-\,\sum _{j=1}^{m}{\frac {\left(m+1-j\right)\,{p}^{j-1}}{\left(q-1\right)^{j}}}$
MGF	For m is odd: $M_{m}\left(t\right)=\sum _{j={\frac {m}{2}}+{\frac {1}{2}}}^{m}{\frac {\left(X_{\it {jm}}+Y_{\it {jm}}\right)~e^{j~t}}{\left(1-e^{t}~q\right)^{j}}}$ For m is even: $M_{m}\left(t\right)=\sum _{j={\frac {m}{2}}}^{m}{\frac {\left(X_{\it {jm}}+Y_{\it {jm}}\right)~e^{j~t}}{\left(1-e^{t}~q\right)^{j}}}$ with $X_{\it {jm}}\,=\,{j-1 \choose 2\,j-m-1}{p}^{m+1-j}\left(1-p-q\right)^{2\,j-m-1}$ and $Y_{\it {jm}}\,=\,{j \choose 2\,j-m}{p}^{-j+m}\left(1-p-q\right)^{2\,j-m}$

Hardy distribution information

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