normal distribution in 2 dimensionsFor a product of two 1-d normal distributions with given mean and standard distribution, we find probability for a given range of pairs of values of the random variables by integrating a double integral over that corresponding region. We set up a single random variable by assigning it a type of distribution with a given mean \316\274 and standard deviation \317\203, then PDF gives us that distribution function. Here is an example of a normal distribution (PDF = Probability Distribution Function). We can take products of 1-d distributions to get useful 2-d distributions which correspond to independent variables.We load the Statistics package and set the PDF function for a given 1-d distribution as in the next section:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNictRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RMCZBcHBseUZ1bmN0aW9uO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHRkIvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRK1N0YXRpc3RpY3NGJ0Y2RjlGQEZDRkNGQEZDRitGQEZDLUY9Ni1RIjpGJ0ZDRkVGR0ZJRktGTUZPRlEvRlRRLDAuMjc3Nzc3OGVtRicvRldGXm9GQEZD1d normal distributionsThe normal distribution is charactreized by a "mean value" \316\274 locating the peak of the distribution and a "standard deviation" \317\203 describing the "spread" of the distribution about the peak. The exponential decays (goes to zero) very quickly because of the square, so its "tail" which extends out to infinity in both directions can be ignored in practice.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 is the plot for a mean of 70 and spread parameter (standard deviation!) of 10: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The pink shaded area under the curve below is the probability that the random variable takes a value between 60 and 80.
You can imagine this to be a test grade distribution where the average grade is 70, and the standard deviation is 10: about 2/3 the students would have gotten a grade in this grade interval if the normal distribution correctly described the class distribution of grades (the area under the curve for this interval is 0.68 very close to 2/3).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[It turns out that there is always a 68 percent probability of being within 1 standard deviation from the mean in any normal distribution. The value of the standard deviation just relables the tickmarks of the "standard distribution" with unit standard deviation.] Explore how the parameters affect the shapeCopying the normal distribution expression and plotting it in a sensible window, then using the Explore command with its two parameters shows very concretely how they affect the distribution 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
SSZzaWdtYUc2Ig==#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot(1/2*sqrt(2)*exp(-1/2*(t-mu)^2/sigma^2)/sqrt(Pi)/sigma,t = -10 .. 10,0 .. 2), ':-Plot0', [':-sigma', ':-mu'], ':-sliders'=["Slidersigma", "Slidermu"], ':-labels'=["Labelsigma", "Labelmu"],':-isplot'=true, ':-handlers'=[sigma=handlertab[sigma],mu=handlertab[mu]]);
#
# End of autogenerated code.
#
SSNtdUc2Ig==#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plot(1/2*sqrt(2)*exp(-1/2*(t-mu)^2/sigma^2)/sqrt(Pi)/sigma,t = -10 .. 10,0 .. 2), ':-Plot0', [':-sigma', ':-mu'], ':-sliders'=["Slidersigma", "Slidermu"], ':-labels'=["Labelsigma", "Labelmu"],':-isplot'=true, ':-handlers'=[sigma=handlertab[sigma],mu=handlertab[mu]]);
#
# End of autogenerated code.
#
2d product normal distributionsNow we take a product of two such distributions, say, two such classes of students with different averages (mean values) and spreads (standard distributions), and ask what the probability is that a pair of students, one in each class, gets a grade within a certain grade interval in each class.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 equals volume under the graph. The actual variables probably have to be nonnegative, so extending the ranges 0..100 in each direction captures nearly all the distribution except for about 0.002 or .2 percent.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjgtRiw2JVEkUERGRidGL0YyLUY2NiQtRiM2Ji1GLDYlUSJYRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZMLyUqc3ltbWV0cmljR0ZMLyUobGFyZ2VvcEdGTC8lLm1vdmFibGVsaW1pdHNHRkwvJSdhY2NlbnRHRkwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ1RidGL0YyRkhGSC1GRTYtUSJ+RidGSEZKL0ZORkxGT0ZRRlNGVUZXRlkvRmduRmVuRjotRjY2JC1GIzYmLUYsNiVRIllGJ0YvRjJGRC1GLDYlUSJ2RidGL0YyRkhGSEZERmluLUZFNi1RIj1GJ0ZIRkpGX29GT0ZRRlNGVUZXL0ZaUSwwLjI3Nzc3NzhlbUYnL0ZnbkZfcC1JI21uR0YkNiRRIjBGJ0ZILUZFNi1RIy4uRidGSEZKRl9vRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJ0Zgby1GYnA2JFEkMTAwRidGSEZERmhvRltwRmFwRmVwRmpwRkQtRiw2JVElYXhlc0YnRi9GMkZbcC1GLDYlUSZib3hlZEYnRi9GMkZIRkgtRkU2LVEiO0YnRkhGSkZNRk9GUUZTRlVGV0ZZRmBwRlxvLUYjNjQtSShtc3Vic3VwR0YkNictRkU2LVErJkludGVncmFsO0YnRkgvRktRJnVuc2V0RicvRk5GX3IvRlBGMS9GUkZfci9GVEYxL0ZWRl9yL0ZYRl9yRllGYG8tRiM2JEZhcEZILUYjNiRGanBGSC8lMXN1cGVyc2NyaXB0c2hpZnRHRmRwLyUvc3Vic2NyaXB0c2hpZnRHRmRwRmhxRjpGPUZcb0Y6RmFvRlxvLUZFNi1RMCZEaWZmZXJlbnRpYWxEO0YnRkhGXnJGYHIvRlBGX3JGYnIvRlRGX3JGZHJGZXJGWUZgb0Zob0Zcb0Zec0ZpbkZjcUZcby1GLDYlUSZldmFsZkYnRi9GMi1GNjYkLUYjNiQtRiw2JVEiJUYnRi9GMkZIRkhGSC1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHRkxGSA==This range of the coordinates below contains 3/4 of the distribution, i.e., the volume under the graph is shown below to be about 0.7505: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 error function "erf" is simply an antiderivative of the normal distribution function, but it cannot be expressed in terms of elementary functions.extra: unit circle in standard deviation normalized variablesIf we introduce new "standard" variables 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 normalized by dividing by their standard deviations, the new variables measure the number of standard deviations from the mean, and a unit circle in these new variables has the form of an ellipse in the original variables: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 fun we plot the part of the probability distribution above this ellipse and then superimpose it on the original distribution in the second plot. The fraction of the whole probability that we calculate looks reasonable.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