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courses:ast201:3 [2023/10/22 05:22] – [3. Probability distributions] asadcourses:ast201:3 [2023/10/22 05:37] (current) – [2. Population and sample] asad
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 ^ Astronomer ^ Precision ^ Accuracy ^ Decision ^ ^ Astronomer ^ Precision ^ Accuracy ^ Decision ^
-| A | Precise | Accurate | Evacuate | +| A (cyan) | Precise | Accurate | Evacuate | 
-| B | Precise | Inaccurate | Stay | +| B (magenta) | Precise | Inaccurate | Stay | 
-| C | Imprecise | Accurate | Uncertain | +| C (yellow) | Imprecise | Accurate | Uncertain | 
-| D | Imprecise | Inaccurate | Uncertain |+| D (black) | Imprecise | Inaccurate | Uncertain |
  
 Systematic errors are more important and dangerous for astronomy than stochastic errors. Astronomers A and B have the same precision, but after all the measurements B realizes that her values differ from the values of everyone else and, hence, unlikely to be true. Another reason for her inaccuracy is that her deviation from others is much greater than her precision. Astronomer C sees a systematic errors in his stochastic errors because his values decrease with time predictably. Systematic errors are more important and dangerous for astronomy than stochastic errors. Astronomers A and B have the same precision, but after all the measurements B realizes that her values differ from the values of everyone else and, hence, unlikely to be true. Another reason for her inaccuracy is that her deviation from others is much greater than her precision. Astronomer C sees a systematic errors in his stochastic errors because his values decrease with time predictably.
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 [[https://colab.research.google.com/drive/1Pc1gv27ywDHavjQiNsZESRY0nij-KHAh?usp=sharing|{{:courses:ast201:stars-iron.png?nolink|}}]] [[https://colab.research.google.com/drive/1Pc1gv27ywDHavjQiNsZESRY0nij-KHAh?usp=sharing|{{:courses:ast201:stars-iron.png?nolink|}}]]
  
-In this example, the velocities of 50 stars perpendicular to the Galactic plane are plotted as a bar-chart histogram. The stars are divided into two sets: 25 stars are iron-rich and the other 25 stars are iron-poor. The iron-poor stars have a greater dispersion of velocities as evident from the plot. The mean and the standard deviation are shown as vertical lines and shades, respectively.+In this example, the velocities of 50 stars perpendicular to the [[uv:mw|Galactic plane]] are plotted as a bar-chart histogram. The stars are divided into two sets: 25 stars are iron-rich and the other 25 stars are iron-poor. The iron-poor stars have a greater dispersion of velocities as evident from the plot. The mean and the standard deviation are shown as vertical lines and shades, respectively.
  
 Standard deviation is sometimes better than variance in describing physical phenomena because σσ has the same unit as μμ, but the units are squared in σ2σ2 statistic. In this example, the variance among iron-rich stars is 57.2557.25 km22 s−2−2 whereas the standard deviation is 7.577.57 km s−1−1. Standard deviation is sometimes better than variance in describing physical phenomena because σσ has the same unit as μμ, but the units are squared in σ2σ2 statistic. In this example, the variance among iron-rich stars is 57.2557.25 km22 s−2−2 whereas the standard deviation is 7.577.57 km s−1−1.
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 Probability distributions can be **continuous** or **discrete**. Continuous distributions allow all integer and fractional values, but the discrete distributions allow only a discrete set of values. For the continuous case, PQ(x)(dx)PQ(x)(dx) is the probability of the value of xx being between xx and x+dxx+dx. And in the discrete case, PQ(xj)PQ(xj) is the probability of the value being xjxj where j=1,2,3,...j=1,2,3,... Probability distributions can be **continuous** or **discrete**. Continuous distributions allow all integer and fractional values, but the discrete distributions allow only a discrete set of values. For the continuous case, PQ(x)(dx)PQ(x)(dx) is the probability of the value of xx being between xx and x+dxx+dx. And in the discrete case, PQ(xj)PQ(xj) is the probability of the value being xjxj where j=1,2,3,...j=1,2,3,...
  
-The probability distributions most used in astronomy are the discrete [[uv:Poisson distribution]] and the continuous [[uv:Gaussian distribution]]. Read the linked //Universe// articles for more about them.+The probability distributions most used in astronomy are the discrete [[uv:poisson|Poisson distribution]] and the continuous [[uv:gaussian|Gaussian distribution]]. Read the linked //Universe// articles for more about them.
  
 Here instead let us compare the two. Poisson only allows non-negative integer values describing the number of events within a duration of time. For example, the number of raindrops falling on a tin-roof in one second, the number of photons falling on the detector of the Chandra X-ray telescope and so on. On the other hand, Gaussian distributions allow any positive or negative value and can describe the multiple measurements of any given quantity. For example, if you measure the magnitude of a star 100 times and get rid of all systematic errors, then the stochastic-error-dominated final results can be described using a Gaussian. Here instead let us compare the two. Poisson only allows non-negative integer values describing the number of events within a duration of time. For example, the number of raindrops falling on a tin-roof in one second, the number of photons falling on the detector of the Chandra X-ray telescope and so on. On the other hand, Gaussian distributions allow any positive or negative value and can describe the multiple measurements of any given quantity. For example, if you measure the magnitude of a star 100 times and get rid of all systematic errors, then the stochastic-error-dominated final results can be described using a Gaussian.
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 and again you see that whether there is a multiplication or division, the fractional errors always add up. The rule of error propagation can be generalized using the following way. and again you see that whether there is a multiplication or division, the fractional errors always add up. The rule of error propagation can be generalized using the following way.
  
-If GG if a function of nn variables and their standard deviations are given by $\sigma,thenthevariancein,thenthevarianceinG$ is given by+If GG if a function of nn variables (xixi) and their standard deviations are given by $\sigma_i,thenthevariancein,thenthevarianceinG$ is given by
  
 σ2=n∑i=1(∂G∂xi)2σ2i+Cσ2=n∑i=1(∂G∂xi)2σ2i+C σ2=n∑i=1(∂G∂xi)2σ2i+Cσ2=n∑i=1(∂G∂xi)2σ2i+C
  
-where CC is the [[uv:covariance]] which we can consider zero for the current purpose. So we have to perform a partial differentiation of the function with respect to each variable xixi and multiply the square of the result with the variance of that variable (σiσi). The rules of subtraction and division shown above can be derived from this.+where CC is the [[uv:covariance]] which we can consider zero for the current purpose. So we have to perform a partial differentiation of the function with respect to each variable xixi and multiply the square of the result with the variance of that variable ($\sigma_i^2$). The rules of propagation for subtraction (addition) and division (multiplication) shown above can be derived from this.
  
courses/ast201/3.1697973739.txt.gz · Last modified: 2023/10/22 05:22 by asad
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