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CHM 421/621 Statistical Mechanics Lecture 7

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  • CHM 421/621 Statistical Mechanics

    Lecture 7

  • Introduction and ReviewLecture Plan

    Review of probability theory (ctd.) • Probability Distributions - Discrete samples

    • Probability Distributions - Continuous samples

    • Averages and variances under a distribution

    • Uncertainty

  • Averages and VariancesAverage

    Consider a property g(x) that depends on the variable (event) x.

    We can compute an average value of g over a series of measurements of x as

    Or, if we know the probability distribution P(x), as

    ḡ =NX

    x

    g(x)P (x)AAACDnicbVDLSsNAFJ34rPUVdelmsBTqpiQi6EYounElFewDmhgm00k6dCYJMxNpCfkCN/6KGxeKuHXtzr9x2mahrQfu5XDOvczc4yeMSmVZ38bS8srq2nppo7y5tb2za+7tt2WcCkxaOGax6PpIEkYj0lJUMdJNBEHcZ6TjD68mfueBCEnj6E6NE+JyFEY0oBgpLXlm1fGRyMIcXkBHptxhlFMlvWyU39/AsDY6hk3dPLNi1a0p4CKxC1IBBZqe+eX0Y5xyEinMkJQ920qUmyGhKGYkLzupJAnCQxSSnqYR4kS62fScHFa10odBLHRFCk7V3xsZ4lKOua8nOVIDOe9NxP+8XqqCczejUZIqEuHZQ0HKoIrhJBvYp4JgxcaaICyo/ivEAyQQVjrBsg7Bnj95kbRP6rZVt29PK43LIo4SOARHoAZscAYa4Bo0QQtg8AiewSt4M56MF+Pd+JiNLhnFzgH4A+PzB6bUmy0=AAACDnicbVDLSsNAFJ34rPUVdelmsBTqpiQi6EYounElFewDmhgm00k6dCYJMxNpCfkCN/6KGxeKuHXtzr9x2mahrQfu5XDOvczc4yeMSmVZ38bS8srq2nppo7y5tb2za+7tt2WcCkxaOGax6PpIEkYj0lJUMdJNBEHcZ6TjD68mfueBCEnj6E6NE+JyFEY0oBgpLXlm1fGRyMIcXkBHptxhlFMlvWyU39/AsDY6hk3dPLNi1a0p4CKxC1IBBZqe+eX0Y5xyEinMkJQ920qUmyGhKGYkLzupJAnCQxSSnqYR4kS62fScHFa10odBLHRFCk7V3xsZ4lKOua8nOVIDOe9NxP+8XqqCczejUZIqEuHZQ0HKoIrhJBvYp4JgxcaaICyo/ivEAyQQVjrBsg7Bnj95kbRP6rZVt29PK43LIo4SOARHoAZscAYa4Bo0QQtg8AiewSt4M56MF+Pd+JiNLhnFzgH4A+PzB6bUmy0=AAACDnicbVDLSsNAFJ34rPUVdelmsBTqpiQi6EYounElFewDmhgm00k6dCYJMxNpCfkCN/6KGxeKuHXtzr9x2mahrQfu5XDOvczc4yeMSmVZ38bS8srq2nppo7y5tb2za+7tt2WcCkxaOGax6PpIEkYj0lJUMdJNBEHcZ6TjD68mfueBCEnj6E6NE+JyFEY0oBgpLXlm1fGRyMIcXkBHptxhlFMlvWyU39/AsDY6hk3dPLNi1a0p4CKxC1IBBZqe+eX0Y5xyEinMkJQ920qUmyGhKGYkLzupJAnCQxSSnqYR4kS62fScHFa10odBLHRFCk7V3xsZ4lKOua8nOVIDOe9NxP+8XqqCczejUZIqEuHZQ0HKoIrhJBvYp4JgxcaaICyo/ivEAyQQVjrBsg7Bnj95kbRP6rZVt29PK43LIo4SOARHoAZscAYa4Bo0QQtg8AiewSt4M56MF+Pd+JiNLhnFzgH4A+PzB6bUmy0=AAACDnicbVDLSsNAFJ34rPUVdelmsBTqpiQi6EYounElFewDmhgm00k6dCYJMxNpCfkCN/6KGxeKuHXtzr9x2mahrQfu5XDOvczc4yeMSmVZ38bS8srq2nppo7y5tb2za+7tt2WcCkxaOGax6PpIEkYj0lJUMdJNBEHcZ6TjD68mfueBCEnj6E6NE+JyFEY0oBgpLXlm1fGRyMIcXkBHptxhlFMlvWyU39/AsDY6hk3dPLNi1a0p4CKxC1IBBZqe+eX0Y5xyEinMkJQ920qUmyGhKGYkLzupJAnCQxSSnqYR4kS62fScHFa10odBLHRFCk7V3xsZ4lKOua8nOVIDOe9NxP+8XqqCczejUZIqEuHZQ0HKoIrhJBvYp4JgxcaaICyo/ivEAyQQVjrBsg7Bnj95kbRP6rZVt29PK43LIo4SOARHoAZscAYa4Bo0QQtg8AiewSt4M56MF+Pd+JiNLhnFzgH4A+PzB6bUmy0=

    ḡ =

    Zg(x)f(x) dx

    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

    Or

    Discrete Continuous

    ḡ =1

    Nexpt

    NexptX

    i=1

    g(xi)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

  • -1 -0.5 0 0.5 1x0

    1

    2

    3

    4

    f(x)

    alpha = 20alpha = 40

    Thus, the smaller the value of

    the fatter the curve

    ↵AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJfROJsmY2ZllZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSgQ31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVqyhpUCaXbERomuGQNy61g7UQzjCPBWtH4dua3npg2XMkHO0lYGONQ8gGnaJ3U7KJIRtgrV/yqPwdZJUFOKpCj3it/dfuKpjGTlgo0phP4iQ0z1JZTwaalbmpYgnSMQ9ZxVGLMTJjNr52SM6f0yUBpV9KSufp7IsPYmEkcuc4Y7cgsezPxP6+T2sF1mHGZpJZJulg0SAWxisxeJ32uGbVi4ghSzd2thI5QI7UuoJILIVh+eZU0L6qBXw3uLyu1mzyOIpzAKZxDAFdQgzuoQwMoPMIzvMKbp7wX7937WLQWvHzmGP7A+/wBi4GPGA==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJfROJsmY2ZllZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSgQ31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVqyhpUCaXbERomuGQNy61g7UQzjCPBWtH4dua3npg2XMkHO0lYGONQ8gGnaJ3U7KJIRtgrV/yqPwdZJUFOKpCj3it/dfuKpjGTlgo0phP4iQ0z1JZTwaalbmpYgnSMQ9ZxVGLMTJjNr52SM6f0yUBpV9KSufp7IsPYmEkcuc4Y7cgsezPxP6+T2sF1mHGZpJZJulg0SAWxisxeJ32uGbVi4ghSzd2thI5QI7UuoJILIVh+eZU0L6qBXw3uLyu1mzyOIpzAKZxDAFdQgzuoQwMoPMIzvMKbp7wX7937WLQWvHzmGP7A+/wBi4GPGA==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJfROJsmY2ZllZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSgQ31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVqyhpUCaXbERomuGQNy61g7UQzjCPBWtH4dua3npg2XMkHO0lYGONQ8gGnaJ3U7KJIRtgrV/yqPwdZJUFOKpCj3it/dfuKpjGTlgo0phP4iQ0z1JZTwaalbmpYgnSMQ9ZxVGLMTJjNr52SM6f0yUBpV9KSufp7IsPYmEkcuc4Y7cgsezPxP6+T2sF1mHGZpJZJulg0SAWxisxeJ32uGbVi4ghSzd2thI5QI7UuoJILIVh+eZU0L6qBXw3uLyu1mzyOIpzAKZxDAFdQgzuoQwMoPMIzvMKbp7wX7937WLQWvHzmGP7A+/wBi4GPGA==AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoMegF48RzAOSJfROJsmY2ZllZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSgQ31ve/vcLa+sbmVnG7tLO7t39QPjxqGpVqyhpUCaXbERomuGQNy61g7UQzjCPBWtH4dua3npg2XMkHO0lYGONQ8gGnaJ3U7KJIRtgrV/yqPwdZJUFOKpCj3it/dfuKpjGTlgo0phP4iQ0z1JZTwaalbmpYgnSMQ9ZxVGLMTJjNr52SM6f0yUBpV9KSufp7IsPYmEkcuc4Y7cgsezPxP6+T2sF1mHGZpJZJulg0SAWxisxeJ32uGbVi4ghSzd2thI5QI7UuoJILIVh+eZU0L6qBXw3uLyu1mzyOIpzAKZxDAFdQgzuoQwMoPMIzvMKbp7wX7937WLQWvHzmGP7A+/wBi4GPGA==

    Averages and VariancesAverageConsider the following distribution function for events

    Gaussian or normal distribution�1 x 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

    x̄ =

    r↵

    1Z

    �1

    x exp��↵x2

    �dx

    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

    x̄2 =

    r↵

    1Z

    �1

    x2 exp��↵x2

    �dx

    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

    = 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

    =1

    2↵AAAB+3icbVBNS8NAEJ34WetXrEcvwSJ4KkkR9CIUvXisYD+gKWWy3bRLN5uwuxFLyF/x4kERr/4Rb/4bt20O2vpg4PHeDDPzgoQzpV3321pb39jc2i7tlHf39g8O7aNKW8WpJLRFYh7LboCKciZoSzPNaTeRFKOA004wuZ35nUcqFYvFg54mtB/hSLCQEdRGGtiVaz+USDIvz+o+8mSM+cCuujV3DmeVeAWpQoHmwP7yhzFJIyo04ahUz3MT3c9QakY4zct+qmiCZIIj2jNUYERVP5vfnjtnRhk6YSxNCe3M1d8TGUZKTaPAdEaox2rZm4n/eb1Uh1f9jIkk1VSQxaIw5Y6OnVkQzpBJSjSfGoJEMnOrQ8ZostAmrrIJwVt+eZW06zXPrXn3F9XGTRFHCU7gFM7Bg0towB00oQUEnuAZXuHNyq0X6936WLSuWcXMMfyB9fkD0KqUSQ==AAAB+3icbVBNS8NAEJ34WetXrEcvwSJ4KkkR9CIUvXisYD+gKWWy3bRLN5uwuxFLyF/x4kERr/4Rb/4bt20O2vpg4PHeDDPzgoQzpV3321pb39jc2i7tlHf39g8O7aNKW8WpJLRFYh7LboCKciZoSzPNaTeRFKOA004wuZ35nUcqFYvFg54mtB/hSLCQEdRGGtiVaz+USDIvz+o+8mSM+cCuujV3DmeVeAWpQoHmwP7yhzFJIyo04ahUz3MT3c9QakY4zct+qmiCZIIj2jNUYERVP5vfnjtnRhk6YSxNCe3M1d8TGUZKTaPAdEaox2rZm4n/eb1Uh1f9jIkk1VSQxaIw5Y6OnVkQzpBJSjSfGoJEMnOrQ8ZostAmrrIJwVt+eZW06zXPrXn3F9XGTRFHCU7gFM7Bg0towB00oQUEnuAZXuHNyq0X6936WLSuWcXMMfyB9fkD0KqUSQ==AAAB+3icbVBNS8NAEJ34WetXrEcvwSJ4KkkR9CIUvXisYD+gKWWy3bRLN5uwuxFLyF/x4kERr/4Rb/4bt20O2vpg4PHeDDPzgoQzpV3321pb39jc2i7tlHf39g8O7aNKW8WpJLRFYh7LboCKciZoSzPNaTeRFKOA004wuZ35nUcqFYvFg54mtB/hSLCQEdRGGtiVaz+USDIvz+o+8mSM+cCuujV3DmeVeAWpQoHmwP7yhzFJIyo04ahUz3MT3c9QakY4zct+qmiCZIIj2jNUYERVP5vfnjtnRhk6YSxNCe3M1d8TGUZKTaPAdEaox2rZm4n/eb1Uh1f9jIkk1VSQxaIw5Y6OnVkQzpBJSjSfGoJEMnOrQ8ZostAmrrIJwVt+eZW06zXPrXn3F9XGTRFHCU7gFM7Bg0towB00oQUEnuAZXuHNyq0X6936WLSuWcXMMfyB9fkD0KqUSQ==AAAB+3icbVBNS8NAEJ34WetXrEcvwSJ4KkkR9CIUvXisYD+gKWWy3bRLN5uwuxFLyF/x4kERr/4Rb/4bt20O2vpg4PHeDDPzgoQzpV3321pb39jc2i7tlHf39g8O7aNKW8WpJLRFYh7LboCKciZoSzPNaTeRFKOA004wuZ35nUcqFYvFg54mtB/hSLCQEdRGGtiVaz+USDIvz+o+8mSM+cCuujV3DmeVeAWpQoHmwP7yhzFJIyo04ahUz3MT3c9QakY4zct+qmiCZIIj2jNUYERVP5vfnjtnRhk6YSxNCe3M1d8TGUZKTaPAdEaox2rZm4n/eb1Uh1f9jIkk1VSQxaIw5Y6OnVkQzpBJSjSfGoJEMnOrQ8ZostAmrrIJwVt+eZW06zXPrXn3F9XGTRFHCU7gFM7Bg0towB00oQUEnuAZXuHNyq0X6936WLSuWcXMMfyB9fkD0KqUSQ==

    f(x) =

    r↵

    ⇡exp

    ��↵x2

    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

  • Averages and VariancesAverageConsider the following distribution function for events

    �1 x 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

    f(x) =

    r↵

    ⇡exp

    ��↵(x� x0)2

    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

    x̄ =

    r↵

    1Z

    �1

    x exp��↵(x� x0)2

    �dx

    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

    = 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

    =1

    2↵+ x20

    AAACA3icbVDLSsNAFJ34rPUVdaebwSIIQkmKoBuh6MZlBfuAJoab6aQdOnkwMxFLCLjxV9y4UMStP+HOv3HaZqGtBy4czrmXe+/xE86ksqxvY2FxaXlltbRWXt/Y3No2d3ZbMk4FoU0S81h0fJCUs4g2FVOcdhJBIfQ5bfvDq7HfvqdCsji6VaOEuiH0IxYwAkpLnrl/4QQCSGbnWc0Bngwgxyf4wbPuap5ZsarWBHie2AWpoAINz/xyejFJQxopwkHKrm0lys1AKEY4zctOKmkCZAh92tU0gpBKN5v8kOMjrfRwEAtdkcIT9fdEBqGUo9DXnSGogZz1xuJ/XjdVwbmbsShJFY3IdFGQcqxiPA4E95igRPGRJkAE07diMgCdidKxlXUI9uzL86RVq9pW1b45rdQvizhK6AAdomNkozNUR9eogZqIoEf0jF7Rm/FkvBjvxse0dcEoZvbQHxifP0Lqlps=AAACA3icbVDLSsNAFJ34rPUVdaebwSIIQkmKoBuh6MZlBfuAJoab6aQdOnkwMxFLCLjxV9y4UMStP+HOv3HaZqGtBy4czrmXe+/xE86ksqxvY2FxaXlltbRWXt/Y3No2d3ZbMk4FoU0S81h0fJCUs4g2FVOcdhJBIfQ5bfvDq7HfvqdCsji6VaOEuiH0IxYwAkpLnrl/4QQCSGbnWc0Bngwgxyf4wbPuap5ZsarWBHie2AWpoAINz/xyejFJQxopwkHKrm0lys1AKEY4zctOKmkCZAh92tU0gpBKN5v8kOMjrfRwEAtdkcIT9fdEBqGUo9DXnSGogZz1xuJ/XjdVwbmbsShJFY3IdFGQcqxiPA4E95igRPGRJkAE07diMgCdidKxlXUI9uzL86RVq9pW1b45rdQvizhK6AAdomNkozNUR9eogZqIoEf0jF7Rm/FkvBjvxse0dcEoZvbQHxifP0Lqlps=AAACA3icbVDLSsNAFJ34rPUVdaebwSIIQkmKoBuh6MZlBfuAJoab6aQdOnkwMxFLCLjxV9y4UMStP+HOv3HaZqGtBy4czrmXe+/xE86ksqxvY2FxaXlltbRWXt/Y3No2d3ZbMk4FoU0S81h0fJCUs4g2FVOcdhJBIfQ5bfvDq7HfvqdCsji6VaOEuiH0IxYwAkpLnrl/4QQCSGbnWc0Bngwgxyf4wbPuap5ZsarWBHie2AWpoAINz/xyejFJQxopwkHKrm0lys1AKEY4zctOKmkCZAh92tU0gpBKN5v8kOMjrfRwEAtdkcIT9fdEBqGUo9DXnSGogZz1xuJ/XjdVwbmbsShJFY3IdFGQcqxiPA4E95igRPGRJkAE07diMgCdidKxlXUI9uzL86RVq9pW1b45rdQvizhK6AAdomNkozNUR9eogZqIoEf0jF7Rm/FkvBjvxse0dcEoZvbQHxifP0Lqlps=AAACA3icbVDLSsNAFJ34rPUVdaebwSIIQkmKoBuh6MZlBfuAJoab6aQdOnkwMxFLCLjxV9y4UMStP+HOv3HaZqGtBy4czrmXe+/xE86ksqxvY2FxaXlltbRWXt/Y3No2d3ZbMk4FoU0S81h0fJCUs4g2FVOcdhJBIfQ5bfvDq7HfvqdCsji6VaOEuiH0IxYwAkpLnrl/4QQCSGbnWc0Bngwgxyf4wbPuap5ZsarWBHie2AWpoAINz/xyejFJQxopwkHKrm0lys1AKEY4zctOKmkCZAh92tU0gpBKN5v8kOMjrfRwEAtdkcIT9fdEBqGUo9DXnSGogZz1xuJ/XjdVwbmbsShJFY3IdFGQcqxiPA4E95igRPGRJkAE07diMgCdidKxlXUI9uzL86RVq9pW1b45rdQvizhK6AAdomNkozNUR9eogZqIoEf0jF7Rm/FkvBjvxse0dcEoZvbQHxifP0Lqlps=

    x2 � x̄2 = 12↵

    AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwY0mKoBuh6MZlBfuApi0300k7dPJgZiItIX/hxl9x40IRt7rzb5y0WWjrgYHDOfdw5x434kwqy/o2Ciura+sbxc3S1vbO7p65f9CUYSwIbZCQh6LtgqScBbShmOK0HQkKvstpyx3fZH7rgQrJwuBeTSPa9WEYMI8RUFrqmxUn1HaWTia9aorPsOOCSCZpr4qvsOMJIImdJlUHeDSCtG+WrYo1A14mdk7KKEe9b345g5DEPg0U4SBlx7Yi1U1AKEY4TUtOLGkEZAxD2tE0AJ/KbjK7K8UnWhlgLxT6BQrP1N+JBHwpp76rJ31QI7noZeJ/XidW3mU3YUEUKxqQ+SIv5liFOCsJD5igRPGpJkAE03/FZAS6C6WrLOkS7MWTl0mzWrGtin13Xq5d53UU0RE6RqfIRheohm5RHTUQQY/oGb2iN+PJeDHejY/5aMHIM4foD4zPH09in18=AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwY0mKoBuh6MZlBfuApi0300k7dPJgZiItIX/hxl9x40IRt7rzb5y0WWjrgYHDOfdw5x434kwqy/o2Ciura+sbxc3S1vbO7p65f9CUYSwIbZCQh6LtgqScBbShmOK0HQkKvstpyx3fZH7rgQrJwuBeTSPa9WEYMI8RUFrqmxUn1HaWTia9aorPsOOCSCZpr4qvsOMJIImdJlUHeDSCtG+WrYo1A14mdk7KKEe9b345g5DEPg0U4SBlx7Yi1U1AKEY4TUtOLGkEZAxD2tE0AJ/KbjK7K8UnWhlgLxT6BQrP1N+JBHwpp76rJ31QI7noZeJ/XidW3mU3YUEUKxqQ+SIv5liFOCsJD5igRPGpJkAE03/FZAS6C6WrLOkS7MWTl0mzWrGtin13Xq5d53UU0RE6RqfIRheohm5RHTUQQY/oGb2iN+PJeDHejY/5aMHIM4foD4zPH09in18=AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwY0mKoBuh6MZlBfuApi0300k7dPJgZiItIX/hxl9x40IRt7rzb5y0WWjrgYHDOfdw5x434kwqy/o2Ciura+sbxc3S1vbO7p65f9CUYSwIbZCQh6LtgqScBbShmOK0HQkKvstpyx3fZH7rgQrJwuBeTSPa9WEYMI8RUFrqmxUn1HaWTia9aorPsOOCSCZpr4qvsOMJIImdJlUHeDSCtG+WrYo1A14mdk7KKEe9b345g5DEPg0U4SBlx7Yi1U1AKEY4TUtOLGkEZAxD2tE0AJ/KbjK7K8UnWhlgLxT6BQrP1N+JBHwpp76rJ31QI7noZeJ/XidW3mU3YUEUKxqQ+SIv5liFOCsJD5igRPGpJkAE03/FZAS6C6WrLOkS7MWTl0mzWrGtin13Xq5d53UU0RE6RqfIRheohm5RHTUQQY/oGb2iN+PJeDHejY/5aMHIM4foD4zPH09in18=AAACF3icbVDLSsNAFJ3UV62vqEs3g0VwY0mKoBuh6MZlBfuApi0300k7dPJgZiItIX/hxl9x40IRt7rzb5y0WWjrgYHDOfdw5x434kwqy/o2Ciura+sbxc3S1vbO7p65f9CUYSwIbZCQh6LtgqScBbShmOK0HQkKvstpyx3fZH7rgQrJwuBeTSPa9WEYMI8RUFrqmxUn1HaWTia9aorPsOOCSCZpr4qvsOMJIImdJlUHeDSCtG+WrYo1A14mdk7KKEe9b345g5DEPg0U4SBlx7Yi1U1AKEY4TUtOLGkEZAxD2tE0AJ/KbjK7K8UnWhlgLxT6BQrP1N+JBHwpp76rJ31QI7noZeJ/XidW3mU3YUEUKxqQ+SIv5liFOCsJD5igRPGpJkAE03/FZAS6C6WrLOkS7MWTl0mzWrGtin13Xq5d53UU0RE6RqfIRheohm5RHTUQQY/oGb2iN+PJeDHejY/5aMHIM4foD4zPH09in18=

    However, note that

    x2 =

    r↵

    1Z

    �1

    x2 exp��↵(x� x0)2

    �dx

    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

  • Averages and VariancesVarianceAverages are not always the only important property to be extracted from a distribution.

    We might also want to know how much the result of a particular experiment would differ from the expected average.

    We can use the variance for this

    Or, if we know the probability distribution P(x), as

    Or

    Discrete Continuous

    � x2 =NX

    x

    (x� x̄)2 P (x)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

    � x2 =

    Z(x� x̄)2f(x) dx

    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

    Why not

    ? (x� x̄)

    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� x2 =1

    Nexpt

    NexptX

    i=1

    (xi � x̄)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

  • Averages and VariancesVariance

    Note that

    Square root of variance is called the standard deviation

    var(x) = � x2 =NX

    x

    (x� x̄)2 P (x) = x2 � x̄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

    �(x) =p

    var(x)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

    What is the standard deviation in the results in the case of rolling a single die?

    Answer: 35/12

  • Averages and VariancesNormalised average valueThe value of std. deviation helps in understanding the meaningfulness of an average and the spread of a distribution.

    To quantify this we can define a normalised average value of x:

    nav(x) =x̄

    �(x)AAACEXicbVC7SgNBFJ31GeMramkzGITYhF0RtBGCNpYRzAOyIdydzCZDZmaXmdmQsOwv2PgrNhaK2NrZ+TdOHoUmHrhwOOde7r0niDnTxnW/nZXVtfWNzdxWfntnd2+/cHBY11GiCK2RiEeqGYCmnElaM8xw2owVBRFw2ggGtxO/MaRKs0g+mHFM2wJ6koWMgLFSp1BKfSWwhGFWGp3ha+yHCkjqB6DSUZalvmY9AdbKOoWiW3anwMvEm5MimqPaKXz53YgkgkpDOGjd8tzYtFNQhhFOs7yfaBoDGUCPtiyVIKhup9OPMnxqlS4OI2VLGjxVf0+kILQei8B2CjB9vehNxP+8VmLCq3bKZJwYKslsUZhwbCI8iQd3maLE8LElQBSzt2LSB5uJsSHmbQje4svLpH5e9tyyd39RrNzM48ihY3SCSshDl6iC7lAV1RBBj+gZvaI358l5cd6dj1nrijOfOUJ/4Hz+ACYonTg=AAACEXicbVC7SgNBFJ31GeMramkzGITYhF0RtBGCNpYRzAOyIdydzCZDZmaXmdmQsOwv2PgrNhaK2NrZ+TdOHoUmHrhwOOde7r0niDnTxnW/nZXVtfWNzdxWfntnd2+/cHBY11GiCK2RiEeqGYCmnElaM8xw2owVBRFw2ggGtxO/MaRKs0g+mHFM2wJ6koWMgLFSp1BKfSWwhGFWGp3ha+yHCkjqB6DSUZalvmY9AdbKOoWiW3anwMvEm5MimqPaKXz53YgkgkpDOGjd8tzYtFNQhhFOs7yfaBoDGUCPtiyVIKhup9OPMnxqlS4OI2VLGjxVf0+kILQei8B2CjB9vehNxP+8VmLCq3bKZJwYKslsUZhwbCI8iQd3maLE8LElQBSzt2LSB5uJsSHmbQje4svLpH5e9tyyd39RrNzM48ihY3SCSshDl6iC7lAV1RBBj+gZvaI358l5cd6dj1nrijOfOUJ/4Hz+ACYonTg=AAACEXicbVC7SgNBFJ31GeMramkzGITYhF0RtBGCNpYRzAOyIdydzCZDZmaXmdmQsOwv2PgrNhaK2NrZ+TdOHoUmHrhwOOde7r0niDnTxnW/nZXVtfWNzdxWfntnd2+/cHBY11GiCK2RiEeqGYCmnElaM8xw2owVBRFw2ggGtxO/MaRKs0g+mHFM2wJ6koWMgLFSp1BKfSWwhGFWGp3ha+yHCkjqB6DSUZalvmY9AdbKOoWiW3anwMvEm5MimqPaKXz53YgkgkpDOGjd8tzYtFNQhhFOs7yfaBoDGUCPtiyVIKhup9OPMnxqlS4OI2VLGjxVf0+kILQei8B2CjB9vehNxP+8VmLCq3bKZJwYKslsUZhwbCI8iQd3maLE8LElQBSzt2LSB5uJsSHmbQje4svLpH5e9tyyd39RrNzM48ihY3SCSshDl6iC7lAV1RBBj+gZvaI358l5cd6dj1nrijOfOUJ/4Hz+ACYonTg=AAACEXicbVC7SgNBFJ31GeMramkzGITYhF0RtBGCNpYRzAOyIdydzCZDZmaXmdmQsOwv2PgrNhaK2NrZ+TdOHoUmHrhwOOde7r0niDnTxnW/nZXVtfWNzdxWfntnd2+/cHBY11GiCK2RiEeqGYCmnElaM8xw2owVBRFw2ggGtxO/MaRKs0g+mHFM2wJ6koWMgLFSp1BKfSWwhGFWGp3ha+yHCkjqB6DSUZalvmY9AdbKOoWiW3anwMvEm5MimqPaKXz53YgkgkpDOGjd8tzYtFNQhhFOs7yfaBoDGUCPtiyVIKhup9OPMnxqlS4OI2VLGjxVf0+kILQei8B2CjB9vehNxP+8VmLCq3bKZJwYKslsUZhwbCI8iQd3maLE8LElQBSzt2LSB5uJsSHmbQje4svLpH5e9tyyd39RrNzM48ihY3SCSshDl6iC7lAV1RBBj+gZvaI358l5cd6dj1nrijOfOUJ/4Hz+ACYonTg=

    nav(x) � 1AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkTccIrdvV526swBaJ25BqlCg2be/vEFM0ogKTThWquc6ifYzLDUjnOZlL1U0wWSMh7RnqMARVX62uD1HF0YZoDCWpoRGC/X3RIYjpWZRYDojrEdq1ZuL/3m9VIe3fsZEkmoqyHJRmHKkYzQPAg2YpETzmSGYSGZuRWSEJSbaxFU2IbirL6+T9lXdderu43W1cVfEUYIzOIcauHADDXiAJrSAwBSe4RXerNx6sd6tj2XrhlXMnMIfWJ8/Z3eTWw==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkTccIrdvV526swBaJ25BqlCg2be/vEFM0ogKTThWquc6ifYzLDUjnOZlL1U0wWSMh7RnqMARVX62uD1HF0YZoDCWpoRGC/X3RIYjpWZRYDojrEdq1ZuL/3m9VIe3fsZEkmoqyHJRmHKkYzQPAg2YpETzmSGYSGZuRWSEJSbaxFU2IbirL6+T9lXdderu43W1cVfEUYIzOIcauHADDXiAJrSAwBSe4RXerNx6sd6tj2XrhlXMnMIfWJ8/Z3eTWw==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkTccIrdvV526swBaJ25BqlCg2be/vEFM0ogKTThWquc6ifYzLDUjnOZlL1U0wWSMh7RnqMARVX62uD1HF0YZoDCWpoRGC/X3RIYjpWZRYDojrEdq1ZuL/3m9VIe3fsZEkmoqyHJRmHKkYzQPAg2YpETzmSGYSGZuRWSEJSbaxFU2IbirL6+T9lXdderu43W1cVfEUYIzOIcauHADDXiAJrSAwBSe4RXerNx6sd6tj2XrhlXMnMIfWJ8/Z3eTWw==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkTccIrdvV526swBaJ25BqlCg2be/vEFM0ogKTThWquc6ifYzLDUjnOZlL1U0wWSMh7RnqMARVX62uD1HF0YZoDCWpoRGC/X3RIYjpWZRYDojrEdq1ZuL/3m9VIe3fsZEkmoqyHJRmHKkYzQPAg2YpETzmSGYSGZuRWSEJSbaxFU2IbirL6+T9lXdderu43W1cVfEUYIzOIcauHADDXiAJrSAwBSe4RXerNx6sd6tj2XrhlXMnMIfWJ8/Z3eTWw==

    nav(x) ⌧ 1AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkcc5cvt21ak7C6B14hakCgWaffvLG8QkjajQhGOleq6TaD/DUjPCaV72UkUTTMZ4SHuGChxR5WeL23N0YZQBCmNpSmi0UH9PZDhSahYFpjPCeqRWvbn4n9dLdXjrZ0wkqaaCLBeFKUc6RvMg0IBJSjSfGYKJZOZWREZYYqJNXGUTgrv68jppX9Vdp+4+Xlcbd0UcJTiDc6iBCzfQgAdoQgsITOEZXuHNyq0X6936WLZuWMXMKfyB9fkDdriTZQ==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkcc5cvt21ak7C6B14hakCgWaffvLG8QkjajQhGOleq6TaD/DUjPCaV72UkUTTMZ4SHuGChxR5WeL23N0YZQBCmNpSmi0UH9PZDhSahYFpjPCeqRWvbn4n9dLdXjrZ0wkqaaCLBeFKUc6RvMg0IBJSjSfGYKJZOZWREZYYqJNXGUTgrv68jppX9Vdp+4+Xlcbd0UcJTiDc6iBCzfQgAdoQgsITOEZXuHNyq0X6936WLZuWMXMKfyB9fkDdriTZQ==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkcc5cvt21ak7C6B14hakCgWaffvLG8QkjajQhGOleq6TaD/DUjPCaV72UkUTTMZ4SHuGChxR5WeL23N0YZQBCmNpSmi0UH9PZDhSahYFpjPCeqRWvbn4n9dLdXjrZ0wkqaaCLBeFKUc6RvMg0IBJSjSfGYKJZOZWREZYYqJNXGUTgrv68jppX9Vdp+4+Xlcbd0UcJTiDc6iBCzfQgAdoQgsITOEZXuHNyq0X6936WLZuWMXMKfyB9fkDdriTZQ==AAAB+3icbVBNS8NAEJ34WetXrEcvi0Wol5KIoMeiF48V7Ac0oWy2m3bpZhN2N6Ul5K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5QcKZ0o7zbW1sbm3v7Jb2yvsHh0fH9kmlreJUEtoiMY9lN8CKciZoSzPNaTeRFEcBp51gfD/3OxMqFYvFk54l1I/wULCQEayN1LcrmScjJPAkr00vkcc5cvt21ak7C6B14hakCgWaffvLG8QkjajQhGOleq6TaD/DUjPCaV72UkUTTMZ4SHuGChxR5WeL23N0YZQBCmNpSmi0UH9PZDhSahYFpjPCeqRWvbn4n9dLdXjrZ0wkqaaCLBeFKUc6RvMg0IBJSjSfGYKJZOZWREZYYqJNXGUTgrv68jppX9Vdp+4+Xlcbd0UcJTiDc6iBCzfQgAdoQgsITOEZXuHNyq0X6936WLZuWMXMKfyB9fkDdriTZQ==

    If then the average has no significance

    If then the average is significant and the

    deviations from average are quite small

    Most probable value✓@f(x)

    @x

    xm

    = 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

    xmAAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnRxEByhL3NJFmyu3fs7onhyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgQ31ve/vcLK6tr6RnGztLW9s7tX3j9omjjVDBssFrFuRdSg4AoblluBrUQjlZHAh2h0PfUfHlEbHqt7O04wlHSgeJ8zap1099SV3XLFr/ozkGUS5KQCOerd8lenF7NUorJMUGPagZ/YMKPaciZwUuqkBhPKRnSAbUcVlWjCbHbqhJw4pUf6sXalLJmpvycyKo0Zy8h1SmqHZtGbiv957dT2L8OMqyS1qNh8UT8VxMZk+jfpcY3MirEjlGnubiVsSDVl1qVTciEEiy8vk+ZZNfCrwe15pXaVx1GEIziGUwjgAmpwA3VoAIMBPMMrvHnCe/HevY95a8HLZw7hD7zPH2agjdw=AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnRxEByhL3NJFmyu3fs7onhyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgQ31ve/vcLK6tr6RnGztLW9s7tX3j9omjjVDBssFrFuRdSg4AoblluBrUQjlZHAh2h0PfUfHlEbHqt7O04wlHSgeJ8zap1099SV3XLFr/ozkGUS5KQCOerd8lenF7NUorJMUGPagZ/YMKPaciZwUuqkBhPKRnSAbUcVlWjCbHbqhJw4pUf6sXalLJmpvycyKo0Zy8h1SmqHZtGbiv957dT2L8OMqyS1qNh8UT8VxMZk+jfpcY3MirEjlGnubiVsSDVl1qVTciEEiy8vk+ZZNfCrwe15pXaVx1GEIziGUwjgAmpwA3VoAIMBPMMrvHnCe/HevY95a8HLZw7hD7zPH2agjdw=AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnRxEByhL3NJFmyu3fs7onhyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgQ31ve/vcLK6tr6RnGztLW9s7tX3j9omjjVDBssFrFuRdSg4AoblluBrUQjlZHAh2h0PfUfHlEbHqt7O04wlHSgeJ8zap1099SV3XLFr/ozkGUS5KQCOerd8lenF7NUorJMUGPagZ/YMKPaciZwUuqkBhPKRnSAbUcVlWjCbHbqhJw4pUf6sXalLJmpvycyKo0Zy8h1SmqHZtGbiv957dT2L8OMqyS1qNh8UT8VxMZk+jfpcY3MirEjlGnubiVsSDVl1qVTciEEiy8vk+ZZNfCrwe15pXaVx1GEIziGUwjgAmpwA3VoAIMBPMMrvHnCe/HevY95a8HLZw7hD7zPH2agjdw=AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnRxEByhL3NJFmyu3fs7onhyE+wsVDE1l9k579xk1yhiQ8GHu/NMDMvSgQ31ve/vcLK6tr6RnGztLW9s7tX3j9omjjVDBssFrFuRdSg4AoblluBrUQjlZHAh2h0PfUfHlEbHqt7O04wlHSgeJ8zap1099SV3XLFr/ozkGUS5KQCOerd8lenF7NUorJMUGPagZ/YMKPaciZwUuqkBhPKRnSAbUcVlWjCbHbqhJw4pUf6sXalLJmpvycyKo0Zy8h1SmqHZtGbiv957dT2L8OMqyS1qNh8UT8VxMZk+jfpcY3MirEjlGnubiVsSDVl1qVTciEEiy8vk+ZZNfCrwe15pXaVx1GEIziGUwjgAmpwA3VoAIMBPMMrvHnCe/HevY95a8HLZw7hD7zPH2agjdw=

    is called the mode of the distribution

  • UncertaintyMotivation

    Rolling a true die vs. biased die

    Case I. All numbers equally likely Pi = 1/6 (Most uncertain)

    Case II. Pi = 1/8 (i=1,5) and P6=3/8 (Lesser uncertainty than I)

    Case III. Pi = 0 (i=1,5) and P6=1 (Least uncertainty)

    Case IV. Pi = 1/8 (i=1,2,4,5,6) and P3=3/8 (Same as II)

    Now consider rolling two dies, one true and one biased as in case III above.

    The uncertainty in this case is the same as the case I. This can also be thought of as a sum of the individual uncertainties.

    In the general case, the total uncertainty is at least that of either case. So the additivity of uncertainties of independent experiments is an appealing property to require.

  • UncertaintyDesirable Properties

    1. The uncertainty of an experiment consisting of two independent experiments should be a sum their individual uncertainties.

    2. The uncertainty should depend on probabilities of all events.

    3. The uncertainty should depend on the probabilities in a symmetric

    fashion.

    4. The least uncertainty should occur when all but one event have

    probability 0. The maximum uncertainty should occur when all events have equal probability.

    H(P1, P2, · · · , PN ) = �NX

    i=1

    Pi ln(Pi)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

    Discrete

    ContinuousH[f ] = �Z

    f(x) ln(f(x)) 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