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For integrating numerically I like to use lambdify and then numpy but therefore the symbols have to be renamed because lambdify does not like backslashes.\u000a\u000a* Numerical evaluation of the first, second, third, and fourth order moments of the triple Gaussian with limits -10 to 10.\u000a\u000a* Added mixture fractions to the Gaussian.\u000a\u000a* Convert results to dataframe for better readability.\u000a\u000a* Check results for third Gaussian, also adjusted the third Gaussian.\u000a\u000a* Tidy up code.\u000a\u000a* Changed name of file.\u000aAdded function for evaluating the fraction w_prime_4_bar divided by w_prime_2_bar_squared which is parameterized.\u000a\u000a* Added columns to the dataframe to check if the numerical evaluation of the 4th order moment is equal to the parameterized evaluation of the 4th order moment.\u000a\u000a* Adjusted symbols and added the constant for lambda_w.\u000a\u000a* Added documentation.\u000a\u000a* Added check for w_prime_4_bar calculated just with the parameters.\u000a\u000a* Improved functions and improved calculating the moments from a dataframe.\u000a\u000a* Tidied up code\u000a\u000a* Tidied up code\u000a\u000a* Reordered document\u000a\u000a* Added some files and restructured the document.\u000a\u000a* Added a file to compute another function\u000a\u000a* Added a file for computing the 4th order moment based on the parameters and checking this with the numeric integrals.\u000a\u000a* fixed notebook\u000a\u000a* Checked some equations analytically.\u000a\u000a* Added document for checking eq_26\u000a\u000a* Added document for checking eq_24\u000a\u000a* Added document for checking eq_33\u000a\u000a* Fixed some parameters\u000a\u000a* finished document for computing the moments numerically.\u000a\u000a* Verified(?) eq 75\u000a\u000a* Changed values for ?verifying? eq 75\u000a\u000a* Adjusted alphas and deltas but did not run it to the end yet.\u000a\u000a* Ran the code with the adjusted alphas and deltas.\u000a\u000a* Calculated the integral based on the marginal of G_3\u000a\u000a* Ran the notebook again.\u000a\u000a* Added Checking for eq 80. Not done yet.\u000a\u000a* Added new document.\u000a\u000a* Calculated w_bar\u000a\u000a* Finished 1 dim integral.\u000a\u000a* Added calculation for sigma_lambda_w and lambda_w.\u000a\u000a* Added calculation for sigma_lambda_w and lambda_w.\u000a\u000a* Added third Gaussian.\u000a\u000a* Calculated 4th order moment.\u000a\u000a* Added files and split up evaluation of the moments for w\u000a\u000a* Verified all moments for w.\u000a\u000a* Fixed a typo.\u000a\u000a* Set up document for another integral.\u000a\u000a* Working on the integral for w_prime_theta_l_prime_bar.\u000a\u000a* Working on the integral for w_prime_theta_l_prime_bar.\u000a\u000a* Did the check for w_prime_theta_l_prime_bar.\u000a\u000a* Did the check for w_prime_theta_l_prime_bar.\u000a\u000a* Did the check for w_prime_theta_l_prime_bar with rational numbers.\u000a\u000a* Add numerical difference.\u000a\u000a* Add numerical difference.\u000a\u000a* Finished the check for w_prime_theta_l_prime\u000a\u000a* Finished the check for w_prime_squared_theta_l_prime\u000a\u000a* Finished the check for theta_l_prime_3_bar\u000a\u000a* Finished the check for w_prime_theta_l_prime_squared_bar\u000a\u000a* Did the checks again with arbitrary distinct values.\u000a\u000a* Did the checks again with arbitrary distinct values.\u000a\u000a* Add check for theta_l_prime_2_bar.\u000a\u000a* Calculated $\\beta$.\u000a\u000a* Corrected a minor mistake.\u000a\u000a* Backed out beta from the theta_prime_l_bar equation.\u000a\u000a* Checked the beta equation numerically.\u000a\u000a* Printed all NaN values.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Reformat Check.\u000a\u000a* Changed Heading.\u000a\u000a* Changed the limits of integration.\u000a\u000a* Delete doc/.ipynb_checkpoints directory\u000a\u000a* Rearrange symbols\u000a\u000a* Added beta to the dataframe.\u000a\u000a* Added diagnostic values to the dataframe.\u000a\u000a* Fixed typo.\u000a\u000a* rearranged dataframe\u000a\u000a* rearranged dataframe\u000a\u000a* rearranged dataframe\u000a\u000a* rearranged dataframe\u000a\u000a* rearranged dataframe\u000a\u000a* Computed the triple integral again.\u000a\u000a* Rearranged symbols.\u000a\u000a* Working on the trivariate Normal.\u000a\u000a* Fixed numerical integration for triple gaussian\u000a\u000a* plot relationship between delta and lambda_w\u000a\u000a* plotted proposed relationship\u000a\u000a* Rearranged files and computed everything again.\u000a\u000a* Added README.md\u000a\u000a* Added file to README.md\u000a\u000a* Add function for plotting\u000a\u000a* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb\u000a\u000a* Finished numerical check for w_prime_q_t_prime_theta_l_prime_bar_beta.ipynb\u000a\u000a* Finished numerical check for theta_l_prime_q_t_prime_bar.ipynb\u000a\u000a* Worked on plotting\u000a\u000a* Worked on README.md\u000a\u000a* replace q_t with r_t\u000a\u000a* replace q_t with r_t\u000a\u000a* replace q_t with r_t\u000a\u000a* Add explanation\u000a\u000a* Reformat code\u000a\u000a* Corrected a file\u000a\u000a* Corrected a file\u000a\u000a* Worked on checking w_prime_r_t_prime_theta_l_prime_bar_beta.ipynb\u000a\u000a* Simplified creating of dataframe.\u000a\u000a* - 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