title ( 'Revenue surface from experimental data' ) plt. ylabel ( 'Change in frequency of email type 2' ) plt. xlabel ( 'Change in frequency of email type 1' ) plt. scatter (,, marker = 'x', label = 'Revenue-maximizing choice' ) plt. subs ( coefficient_values ) y_star = critical_points. colorbar ( label = 'Revenue per user' ) x_star = critical_points. PercentFormatter ( 1 )) x_y_pairs = cartesian () z = x_plot, y_plot = zip ( * x_y_pairs ) plt. We can now learn where the maxima of the function are, doing some basic calculus. However, computing closed-form expressions for the above gives us some intuition about how the distribution behaves more generally, and could be the starting point for further analysis like computing the standard errors of $s$. Scipy lets us do all of these numerically (using functions like mean(), var(), and fit(data)). How might we estimate $s$ from some data? If we knew the relationship between the first moment and $s$, we could use the Method of Moments for this univariate distribution.What are the moments of this distribution? How do the mean and variance of the distribution depend on $s$?.If we’re going to use this distribution, there are a few questions we’d like to answer about it: The scipy version is implemented in terms of a scale parameter which we’ll call $s$. This is implemented in scipy as halfnorm. Symbolic Integration: Finding the moments of a probability distributionĪ simple model for a continuous, non-negative random variable is a half-normal distribution. Then, we’ll find the maximum of a model by finding its partial derivatives symbolically, and setting it to zero. We’ll start by computing the expected value of a distribution by doing a symbolic definite integral. Here are two examples of recent places I’ve used Sympy to do calculus. In those cases, I’m thankful to be able to check my work and make it reproducible with Sympy, a symbolic mathematics library in Python. My job seems to involve just enough calculus that I can’t afford to forget it, but little enough that I always feel rusty when I need to do it. Symbolic Calculus in Python: Simple Samples of Sympy
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