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!‹ʉ
7‹ʉ 
7‹ʉ …‹
Á‰ ›‹Á‰  ›‹Á‰ é‹ ¸‰ ÿ‹¸‰  ÿ‹¸‰
 ¯‰
¯‰ 
¯‰ ±Œ
¦‰ ÇŒ¦‰  ÇŒ¦‰ + + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mg‰ƒg‰ ƒg‰Ñ^‰ç^‰ ç^‰5^‰ñ^‰ ^‰g‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰•°g‰•°^‰^‰{ _ }Ág‰p‰y‰‚‰‹‰”‰¦‰¯‰ ¸‰
Á‰ ʉ Ó‰
܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ÓŠÓ‰ ÓŠÓ‰
!‹ʉ
7‹ʉ 
7‹ʉ …‹
Á‰ ›‹Á‰  ›‹Á‰ é‹ ¸‰ ÿ‹¸‰  ÿ‹¸‰
 ¯‰
¯‰ 
¯‰ ±Œ
¦‰ ÇŒ¦‰  ÇŒ¦‰ + + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mg‰ƒg‰ ƒg‰Ñ^‰ç^‰ ç^‰5^‰ô^‰ ^‰g‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰•°g‰•°^‰^‰{ _ }Áp‰y‰‚‰‹‰”‰¦‰¯‰¸‰ Á‰
ʉ Ӊ ܉
å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉
¾ŠÓ‰
ÓŠÓ‰ 
ÓŠÓ‰ !‹
ʉ 7‹ʉ  7‹ʉ …‹ Á‰ ›‹Á‰  ›‹Á‰
é‹ ¸‰
ÿ‹¸‰ 
ÿ‹¸‰ 
¯‰ ¯‰  ¯‰±Œ ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mg‰ƒg‰ ƒg‰Ñg‰ëg‰g‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰•°g‰g‰{ _ }Áp‰y‰‚‰‹‰”‰¦‰¯‰¸‰ Á‰
ʉ Ӊ ܉
å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉
¾ŠÓ‰
ÓŠÓ‰ 
ÓŠÓ‰ !‹
ʉ 7‹ʉ  7‹ʉ …‹ Á‰ ›‹Á‰  ›‹Á‰
é‹ ¸‰
ÿ‹¸‰ 
ÿ‹¸‰ 
¯‰ ¯‰  ¯‰±Œ ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mg‰ƒg‰ ƒg‰Ñg‰îg‰g‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰•°g‰g‰{ _ }Áy‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ ʉ
Ó‰ ܉ å‰
å‰÷‰  Šå‰ ÷‰  Šå‰
܉
܉ 
܉ ¾Š
Ó‰ ÓŠÓ‰  ÓŠÓ‰ !‹ ʉ 7‹ʉ  7‹ʉ
…‹ Á‰
›‹Á‰ 
›‹Á‰ é‹
¸‰ ÿ‹¸‰  ÿ‹¸‰ ¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mp‰åp‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰p‰{ _ }Áy‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ ʉ
Ó‰ ܉ å‰
å‰÷‰  Šå‰ ÷‰  Šå‰
܉
܉ 
܉ ¾Š
Ó‰ ÓŠÓ‰  ÓŠÓ‰ !‹ ʉ 7‹ʉ  7‹ʉ
…‹ Á‰
›‹Á‰ 
›‹Á‰ é‹
¸‰ ÿ‹¸‰  ÿ‹¸‰ ¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ p‰p‰ p‰mp‰èp‰p‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰•°p‰p‰{ _ }Á
‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉ Ó‰
܉ å‰ å‰
÷‰  Šå‰ 
÷‰  Šå‰ 
܉ ܉  ܉ ¾Š Ó‰ ÓŠÓ‰  ÓŠÓ‰
!‹ ʉ
7‹ʉ 
7‹ʉ …‹
Á‰ ›‹Á‰  ›‹Á‰é‹ ¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ y‰ßy‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰y‰{ _ }Á
‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉ Ó‰
܉ å‰ å‰
÷‰  Šå‰ 
÷‰  Šå‰ 
܉ ܉  ܉ ¾Š Ó‰ ÓŠÓ‰  ÓŠÓ‰
!‹ ʉ
7‹ʉ 
7‹ʉ …‹
Á‰ ›‹Á‰  ›‹Á‰é‹ ¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Žy‰»Žy‰ »Ž
y‰ y‰ây‰y‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰•°y‰y‰{ _ }Á
‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰ ܉
å‰ å‰ ÷‰  Šå‰  ÷‰  Šå‰  ܉
 ܉ 
 ܉

¾Š Ó‰
ÓŠÓ‰ 
ÓŠÓ‰ !‹
ʉ 7‹ʉ  7‹ʉ…‹ Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Ž‚‰Ù‚‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰‚‰{ _ }Á
‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰ ܉
å‰ å‰ ÷‰  Šå‰  ÷‰  Šå‰  ܉
 ܉ 
 ܉

¾Š Ó‰
ÓŠÓ‰ 
ÓŠÓ‰ !‹
ʉ 7‹ʉ  7‹ʉ…‹ Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‚‰‚‰  ‚‰¥Ž‚‰Ü‚‰‚‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰•°‚‰‚‰{ _ }Á ”‰¦‰¯‰¸‰Á‰ʉÓ‰܉ å‰
å‰ ÷‰  Šå‰  ÷‰  Šå‰
 ܉ 
܉  
܉  ¾Š
Ó‰
 ÓŠÓ‰ 
 ÓŠÓ‰
!‹ ʉ7‹ʉ 7‹ʉ…‹Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‹‰Ó‹‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰‹‰{ _ }Á ”‰¦‰¯‰¸‰Á‰ʉÓ‰܉ å‰
å‰ ÷‰  Šå‰  ÷‰  Šå‰
 ܉ 
܉  
܉  ¾Š
Ó‰
 ÓŠÓ‰ 
 ÓŠÓ‰
!‹ ʉ7‹ʉ 7‹ʉ…‹Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý‹‰ó‹‰ ó ‹‰‹‰Ö‹‰‹‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰•°‹‰‹‰{ _ }Á
¦‰¯‰¸‰Á‰ʉÓ‰܉å‰ å‰
÷‰  Šå‰ 
÷‰  Šå‰ 
܉  ܉   ܉ ¾Š Ó‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ
7‹ʉ 
7‹ʉ
…‹Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý”‰Í”‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰”‰{ _ }Á
¦‰¯‰¸‰Á‰ʉÓ‰܉å‰ å‰
÷‰  Šå‰ 
÷‰  Šå‰ 
܉  ܉   ܉ ¾Š Ó‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ
7‹ʉ 
7‹ʉ
…‹Á‰›‹Á‰ ›‹Á‰é‹¸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + y”‰”‰ 
”‰Ý”‰Ð”‰”‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°•°”‰”‰{ _ }Á
¦‰¯‰¸‰Á‰ʉÓ‰܉å‰å‰ ÷‰  Šå‰  ÷‰  Šå‰ ܉
܉ 
܉
¾ŠÓ‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ 7‹ʉ  7‹ʉ …‹Á‰
›‹Á‰ 
›‹Á‰
鋸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + yǦ‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°{ _ }Á
¦‰¯‰¸‰Á‰ʉÓ‰܉å‰å‰ ÷‰  Šå‰  ÷‰  Šå‰ ܉
܉ 
܉
¾ŠÓ‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ 7‹ʉ  7‹ʉ …‹Á‰
›‹Á‰ 
›‹Á‰
鋸‰ÿ‹¸‰ ÿ‹¸‰¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰+ + yʦ‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰•°{ _ }Á ¯‰¸‰Á‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉ ܉  ܉ ¾ŠÓ‰
ÓŠÓ‰ 
ÓŠÓ‰
!‹ʉ 7‹ʉ  7‹ʉ …‹Á‰ ›‹Á‰  ›‹Á‰ 鋸‰
ÿ‹¸‰ 
ÿ‹¸‰
¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰¦‰Á¦‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰¦‰{ _ }Á ¯‰¸‰Á‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉ ܉  ܉ ¾ŠÓ‰
ÓŠÓ‰ 
ÓŠÓ‰
!‹ʉ 7‹ʉ  7‹ʉ …‹Á‰ ›‹Á‰  ›‹Á‰ 鋸‰
ÿ‹¸‰ 
ÿ‹¸‰
¯‰¯‰ ¯‰±Œ¦‰ÇŒ¦‰ ÇŒ¦‰¦‰Ä¦‰¦‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰•°¦‰¦‰{ _ }Á¸‰Á‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ
7‹ʉ 
7‹ʉ
…‹Á‰ ›‹Á‰  ›‹Á‰ 鋸‰ ÿ‹¸‰  ÿ‹¸‰ ¯‰
¯‰ 
¯‰
±Œ¯‰»¯‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰¯‰{ _ }Á¸‰Á‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ ÓŠÓ‰  ÓŠÓ‰ !‹ʉ
7‹ʉ 
7‹ʉ
…‹Á‰ ›‹Á‰  ›‹Á‰ 鋸‰ ÿ‹¸‰  ÿ‹¸‰ ¯‰
¯‰ 
¯‰
±Œ¯‰¾¯‰¯‰¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰•°¯‰¯‰{ _ }ÁÁ‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ÓŠÓ‰ ÓŠÓ‰!‹ʉ 7‹ʉ  7‹ʉ …‹Á‰
›‹Á‰ 
›‹Á‰
鋸‰ ÿ‹¸‰  ÿ‹¸‰ ¸‰µ¸‰ ¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰¸‰{ _ }ÁÁ‰ʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ÓŠÓ‰ ÓŠÓ‰!‹ʉ 7‹ʉ  7‹ʉ …‹Á‰
›‹Á‰ 
›‹Á‰
鋸‰ ÿ‹¸‰  ÿ‹¸‰ ¸‰¸¸‰ ¸‰Á‰ʉÓ‰܉剕°å‰•°܉•°Ó‰•°ʉ•°Á‰•°¸‰¸‰{ _ }ÁʉÓ‰܉å‰å‰÷‰  Šå‰ ÷‰  Šå‰܉܉ ܉¾ŠÓ‰ÓŠÓ‰ ÓŠÓ‰!‹ʉ7‹ʉ 7‹ʉ…‹Á‰ ›‹Á‰  ›‹Á‰ é‹Á‰¯Á‰
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ü˜ô+ü˜*' Utilities for random number generationÁÃúüÇJG Rand provides utilities to generate random numbers, to convert them toÁü’KH useful types and distributions, and some randomness-related algorithms.ÁÞúŒâ # Quick StartÁôúüøHE To get you started quickly, the easiest and highest-level way to getÁüÁDA a random value is to use [`random()`]; alternatively you can useÁü†PM [`thread_rng()`]. The [`Rng`] trait provides a useful API on all RNGs, whileÁü×=: the [`distributions`] and [`seq`] modules provide furtherÁü•! functionality on top of RNGs.Á·ú ```ÁÌà use rand::prelude::*;ÁÝúüá.+ if rand::random() { // generates a booleanÁüJG // Try printing a random unicode code point (probably a bad idea)!ÁüÛ52 println!("char: {}", rand::random::<char>());Á, úü› %" let mut rng = rand::thread_rng();ÁüÁ @= let y: f64 = rng.gen(); // generates a float between 0 and 1Á
úü†
0- let mut nums: Vec<i32> = (1..100).collect();ÁÜ·
 nums.shuffle(&mut rng);Á
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 # The BookÁî
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=: For the user guide and further documentation, please readÁü° =: [The Rust Rand Book](https://rust-random.github.io/book).Á%https://www.rust-lang.org/favicon.icoÁ„Ì 9https://www.rust-lang.org/logos/rust-logo-128x128-blk.pngÁlû #https://rust-random.github.io/rand/ÁlŒ
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 ”» øûìïÛÞÅȬ½ü˜<9 Generating random samples from probability distributionsÁÕúüÙLI This module is the home of the [`Distribution`] trait and several of itsÁü¦FC implementations. It is the workhorse behind some of the convenientÁüíGD functionality of the [`Rng`] trait, e.g. [`Rng::gen`] and of courseÁ¤µ [`Rng::sample`].ÁÊúüÎIF Abstractly, a [probability distribution] describes the probability ofÁü˜1. occurrence of each value in its sample space.ÁÊúüÎNK More concretely, an implementation of `Distribution<T>` for type `X` is anÁüIF algorithm for choosing values from the sample space (a subset of `T`)ÁüçMJ according to the distribution `X` represents, using an external source ofÁüµ:7 randomness (an RNG supplied to the `sample` function).ÁðúüôFC A type `X` may implement `Distribution<T>` for multiple types `T`.Áü» IF Any type implementing [`Distribution`] is stateless (i.e. immutable),Áü…
NK but it may have internal parameters set at construction time (for example,ÁüÔ
PM [`Uniform`] allows specification of its sample space as a range within `T`).Á¥ ú© úü­ ! # The `Standard` distributionÁÏ úüÓ FC The [`Standard`] distribution is important to mention. This is theÁüš IF distribution used by [`Rng::gen`] and represents the "default" way toÁüä MJ produce a random value for many different types, including most primitiveÁü²
LI types, tuples, arrays, and a few derived types. See the documentation ofÁüÿ
" [`Standard`] for more details.Á¢úü¦OL Implementing `Distribution<T>` for [`Standard`] for user types `T` makes itÁüöJG possible to generate type `T` with [`Rng::gen`], and by extension alsoÁüÁ! with the [`random`] function.ÁãúÄç ## Random charactersÁúü„JG [`Alphanumeric`] is a simple distribution to sample random letters andÁüÏMJ numbers of the `char` type; in contrast [`Standard`] may sample any validÁ\ `char`.Á©ú­úä± # Uniform numeric rangesÁÎúüÒMJ The [`Uniform`] distribution is more flexible than [`Standard`], but alsoÁü KH more specialised: it supports fewer target types, but allows the sampleÁüìKH space to be specified as an arbitrary range within its target type `T`.Áü¸NK Both [`Standard`] and [`Uniform`] are in some sense uniform distributions.Áúü‹NK Values may be sampled from this distribution using `Rng::sample(Range)` orÁüÚ<9 by creating a distribution object with [`Uniform::new`],Áü—NK [`Uniform::new_inclusive`] or `From<Range>`. When the range limits are notÁüæEB known at compile time it is typically faster to reuse an existingÁü¬74 `Uniform` object than to call `Rng::sample(Range)`.ÁäúüèHE User types `T` may also implement `Distribution<T>` for [`Uniform`],Áü±HE although this is less straightforward than for [`Standard`] (see theÁüúLI documentation in the [`uniform`] module). Doing so enables generation ofÁüÇ2/ values of type `T` with `Rng::sample(Range)`.Áúúüþ  ## Open and half-open rangesÁŸúü£KH There are surprisingly many ways to uniformly generate random floats. AÁüïLI range between 0 and 1 is standard, but the exact bounds (open vs closed)Áü¼QN and accuracy differ. In addition to the [`Standard`] distribution Rand offersÁüŽSP [`Open01`] and [`OpenClosed01`]. See "Floating point implementation" section ofÁüâ0- [`Standard`] documentation for more details.ÁúÔ— # Non-uniform samplingÁ²úü¶MJ Sampling a simple true/false outcome with a given probability has a name:Áü„GD the [`Bernoulli`] distribution (this is used by [`Rng::gen_bool`]).ÁÌúüÐEB For weighted sampling from a sequence of discrete values, use theÁü–# [`WeightedIndex`] distribution.Áºúü¾JG This crate no longer includes other non-uniform distributions; insteadÁü‰GD it is recommended that you use either [`rand_distr`] or [`statrs`].ÁÑúÕúüÙVS [probability distribution]: https://en.wikipedia.org/wiki/Probability_distributionÁü° 74 [`rand_distr`]: https://crates.io/crates/rand_distrÁüè /, [`statrs`]: https://crates.io/crates/statrsÁì™! [`random`]: crate::randomÁü·!7ÈŠüï!/‰‹ FQqšªËæèÔµ
¬FL‰&âùFt”&ŸãdÂ& åéGDÐ&­æØH<Ú&¹çTš'Aè†J4¼'Ïê†JÐêÎIdÄ'ÇëÎIÈëÒMdè'ìÒMìÙ,(Õ í²e<¸(¼
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± ´ü‘+»À|ÿ¼ù«¯º®üÖ7üÖJG Construct a new `Bernoulli` with the given probability of success `p`.Á¥ú|­ # PrecisionÁÁúüÉHE For `p = 1.0`, the resulting distribution will always generate true.Áü–IF For `p = 0.0`, the resulting distribution will always generate false.ÁäúüìLI This method is accurate for any input `p` in the range `[0, 1]` which isÁü½B? a multiple of 2<sup>-64</sup>. (Note that not all multiples ofÁü„?< 2<sup>-64</sup> in `[0, 1]` can be represented as a `f64`.)ÁÝÖ¿Ù¿Ú¿«Û¿ˆÜ¿Ý¿Þ¿ˆ‚†f|Â
º®«¯
» áüÌ"XüµB? Construct a new `Bernoulli` with the probability of success ofÁüüDA `numerator`-in-`denominator`. I.e. `new_ratio(2, 3)` will returnÁüÅJG a `Bernoulli` with a 2-in-3 chance, or about 67%, of returning `true`.Áúüœ EB return `true`. If `numerator == 0` it will always return `false`.Áüæ IF For `numerator > denominator` and `denominator == 0`, this returns anÁü´!HE error. Otherwise, for `numerator == denominator`, samples are alwaysÁü"85 true; for `numerator == 0` samples are always false.ÁTÓ"³˜
 numeratorÁLÞ" denominatorÁ\î"üŽ&%º®"üÈ&64Ë&ÉÊ Éº® ÊïÝ£ ã&ÎìïÝ Ò&ïÝÕ&
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¹½,ÊŽùŒ5ßúH>TÓ-¬ý Ì ü•NK Types (distributions) that can be used to create a random instance of `T`.ÁäúüèA> It is possible to sample from a distribution through both theÁüªGD `Distribution` and [`Rng`] traits, via `distr.sample(&mut rng)` andÁüòOL `rng.sample(distr)`. They also both offer the [`sample_iter`] method, whichÁüÂ<9 produces an iterator that samples from the distribution.ÁÿúüƒNK All implementations are expected to be immutable; this has the significantÁüÒDA advantage of not needing to consider thread safety, and for mostÁü—IF distributions efficient state-less sampling algorithms are available.ÁáúüåKH Implementations are typically expected to be portable with reproducibleÁü± :7 results when used with a PRNG with fixed seed; see theÁüì NK [portability chapter](https://rust-random.github.io/book/portability.html)Áü»
NK of The Rust Rand Book. In some cases this does not apply, e.g. the `usize`ÁüŠ C@ type requires different sampling on 32-bit and 64-bit machines.ÁÎ úüÒ .+ [`sample_iter`]: Distribution::sample_iterÁd  ¥!! Îìü ïÍìΰ ˜  ª*ª0"$&"$&"&$24üò 4ü¡ LI Generate a random value of `T`, using `rng` as the source of randomness.Á4õ ËÌ Ë•ª Ì£ΰ £##
 Î섬 ü „¬ÿ 
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 the source of randomness.Áúü¥C@ Note that this function takes `self` by value. This works sinceÁüíDA `Distribution<T>` is impl'd for `&D` where `D: Distribution<T>`,Áü¶IF however borrowing is not automatic hence `distr.sample_iter(...)` mayÁü„EB need to be replaced with `(&distr).sample_iter(...)` to borrow orÁüÎC@ `(&*distr).sample_iter(...)` to reborrow an existing reference.Áú
# ExampleÁ°úÑÞÌÄ use rand::thread_rng;ÁüâMJ use rand::distributions::{Distribution, Alphanumeric, Uniform, Standard};Á´úü¼ let mut rng = thread_rng();Áàú¼è // Vec of 16 x f32:Áü„HE let v: Vec<f32> = Standard.sample_iter(&mut rng).take(16).collect();ÁÑú // String:Áüì  let s: String = AlphanumericÁô‘ .sample_iter(&mut rng)Á„´
.take(7)ÁÄÉ .map(char::from)Áœæ .collect();Áþú¤† // Dice-rolling:ÁüŸ1. let die_range = Uniform::new_inclusive(1, 6);ÁüÕ74 let mut roll_die = die_range.sample_iter(&mut rng);Áü‘)& while roll_die.next().unwrap() != 6 {Áü¿,) println!("Not a 6; rolling again!");ÁÚàÑÞ\‰•ª„¬­­éG±‘H²ì+³¦HQ[žWrŠ•ª„¬ΰ £%% Í섬 ¿¬ÓÍ앪
$•ªΰ„¬$˜ì+ž<>üþkü÷JG Create a distribution of values of 'S' by mapping the output of `Self`ÁÜÆ through the closure `F`Áæú¥°úÑÞ̔԰ü²52 use rand::distributions::{Distribution, Uniform};ÁìúüôÚ±˜úü +( let die = Uniform::new_inclusive(1, 6);ÁüÐ2/ let even_number = die.map(|num| num % 2 == 0);Áü‡)& while !even_number.sample(&mut rng) {Áüµ.+ println!("Still odd; rolling again!");ÁÚàÑÞ•ª¹¹ØH¾‘H¿IÀ¦HhúºM˜D•ªƒ¾ΰž² '¤('( Í샾 Íìž² ˆ"ƒ¾ΰƒ¾±¿˜"ž² Ò¹
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?ž±”²ª±ž²ÆÊ$Ñ-ì+×-;=¤ß/¤ . `String` samplerÁµ.úü¹.PM Sampling a `String` of random characters is not quite the same as collectingÁüŠ/:7 a sequence of chars. This trait contains some helpers.Á»¤ŒË/ÌÅ/Té/AAªüß/–AªªBDBDDBü¨0Wüú/)& Append `len` random chars to `string`Ál«0ÓÔÕ Ó•ª Ô¤± ÕúHúHßûHÁ‘4Ý­f‰úM÷A£CC Ê0AÎ줱 ¹0ßü0
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