How to use the reliability.Distributions.Lognormal_Distribution function in reliability

# creates a distribution object of the best fitting distribution and assigns its name
        best_dist = df3.index.values[0]
        self.best_distribution_name = best_dist
        if best_dist == 'Weibull_2P':
            self.best_distribution = Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta)
        elif best_dist == 'Weibull_3P':
            self.best_distribution = Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma)
        elif best_dist == 'Gamma_2P':
            self.best_distribution = Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta)
        elif best_dist == 'Gamma_3P':
            self.best_distribution = Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma)
        elif best_dist == 'Lognormal_2P':
            self.best_distribution = Lognormal_Distribution(mu=self.Lognormal_2P_mu, sigma=self.Lognormal_2P_sigma)
        elif best_dist == 'Lognormal_3P':
            self.best_distribution = Lognormal_Distribution(mu=self.Lognormal_3P_mu, sigma=self.Lognormal_3P_sigma, gamma=self.Lognormal_3P_gamma)
        elif best_dist == 'Exponential_1P':
            self.best_distribution = Exponential_Distribution(Lambda=self.Expon_1P_lambda)
        elif best_dist == 'Exponential_2P':
            self.best_distribution = Exponential_Distribution(Lambda=self.Expon_2P_lambda, gamma=self.Expon_2P_gamma)
        elif best_dist == 'Normal_2P':
            self.best_distribution = Normal_Distribution(mu=self.Normal_2P_mu, sigma=self.Normal_2P_sigma)
        elif best_dist == 'Beta_2P':
            self.best_distribution = Beta_Distribution(alpha=self.Beta_2P_alpha, beta=self.Beta_2P_beta)

        # print the results
        if print_results is True:  # printing occurs by default
            pd.set_option('display.width', 200)  # prevents wrapping after default 80 characters
            pd.set_option('display.max_columns', 9)  # shows the dataframe without ... truncation
            print(self.results)

        if show_histogram_plot is True:

self.success = False
            warnings.warn('Fitting using Autograd FAILED for Lognormal_2P. The fit from Scipy was used instead so results may not be accurate.')
            self.mu = np.log(sp[2])
            self.sigma = sp[0]

        params = [self.mu, self.sigma]
        k = len(params)
        n = len(all_data)
        LL2 = 2 * Fit_Lognormal_2P.LL(params, failures, right_censored)
        self.loglik2 = LL2
        if n - k - 1 > 0:
            self.AICc = 2 * k + LL2 + (2 * k ** 2 + 2 * k) / (n - k - 1)
        else:
            self.AICc = 'Insufficient data'
        self.BIC = np.log(n) * k + LL2
        self.distribution = Lognormal_Distribution(mu=self.mu, sigma=self.sigma)

        # confidence interval estimates of parameters
        Z = -ss.norm.ppf((1 - CI) / 2)
        if force_sigma is None:
            hessian_matrix = hessian(Fit_Lognormal_2P.LL)(np.array(tuple(params)), np.array(tuple(failures)), np.array(tuple(right_censored)))
            covariance_matrix = np.linalg.inv(hessian_matrix)
            self.mu_SE = abs(covariance_matrix[0][0]) ** 0.5
            self.sigma_SE = abs(covariance_matrix[1][1]) ** 0.5
            self.Cov_mu_sigma = abs(covariance_matrix[0][1])
            self.mu_upper = self.mu + (Z * self.mu_SE)  # these are unique to normal and lognormal mu params
            self.mu_lower = self.mu + (-Z * self.mu_SE)
            self.sigma_upper = self.sigma * (np.exp(Z * (self.sigma_SE / self.sigma)))
            self.sigma_lower = self.sigma * (np.exp(-Z * (self.sigma_SE / self.sigma)))
        else:
            hessian_matrix = hessian(Fit_Lognormal_2P.LL_fs)(np.array(tuple([self.mu])), np.array(tuple(failures)), np.array(tuple(right_censored)), np.array(tuple([force_sigma])))
            covariance_matrix = np.linalg.inv(hessian_matrix)

plt.legend()
        plt.xlim([xmin, xmax])
        plt.title('Probability Density Function')
        plt.xlabel('Data')
        plt.ylabel('Probability density')
        plt.legend()

        plt.subplot(122)  # CDF
        plt.bar(center, hist_cumulative * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')
        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')
        Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta , \gamma$)')
        Exponential_Distribution(Lambda=self.Expon_1P_lambda).CDF(xvals=xvals, label=r'Exponential ($\lambda$)')
        Exponential_Distribution(Lambda=self.Expon_2P_lambda, gamma=self.Expon_2P_gamma).CDF(xvals=xvals, label=r'Exponential ($\lambda , \gamma$)')
        Lognormal_Distribution(mu=self.Lognormal_2P_mu, sigma=self.Lognormal_2P_sigma).CDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma$)')
        Lognormal_Distribution(mu=self.Lognormal_3P_mu, sigma=self.Lognormal_3P_sigma, gamma=self.Lognormal_3P_gamma).CDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma , \gamma$)')
        Normal_Distribution(mu=self.Normal_2P_mu, sigma=self.Normal_2P_sigma).CDF(xvals=xvals, label=r'Normal ($\mu , \sigma$)')
        if max(X) <= 1:  # condition for Beta Dist to be fitted
            Beta_Distribution(alpha=self.Beta_2P_alpha, beta=self.Beta_2P_beta).CDF(xvals=xvals, label=r'Beta ($\alpha , \beta$)')
        plt.legend()
        plt.xlim([xmin, xmax])
        plt.title('Cumulative Distribution Function')
        plt.xlabel('Data')
        plt.ylabel('Cumulative probability density')
        plt.suptitle('Histogram plot of each fitted distribution')
        plt.legend()

# make this histogram. Can't use plt.hist due to need to scale the heights when there's censored data
        num_bins = min(int(len(X) / 2), 30)
        hist, bins = np.histogram(X, bins=num_bins, density=True)
        hist_cumulative = np.cumsum(hist) / sum(hist)
        width = np.diff(bins)
        center = (bins[:-1] + bins[1:]) / 2
        plt.bar(center, hist * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')

        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).PDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).PDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).PDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')
        Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma).PDF(xvals=xvals, label=r'Gamma ($\alpha , \beta , \gamma$)')
        Exponential_Distribution(Lambda=self.Expon_1P_lambda).PDF(xvals=xvals, label=r'Exponential ($\lambda$)')
        Exponential_Distribution(Lambda=self.Expon_2P_lambda, gamma=self.Expon_2P_gamma).PDF(xvals=xvals, label=r'Exponential ($\lambda , \gamma$)')
        Lognormal_Distribution(mu=self.Lognormal_2P_mu, sigma=self.Lognormal_2P_sigma).PDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma$)')
        Lognormal_Distribution(mu=self.Lognormal_3P_mu, sigma=self.Lognormal_3P_sigma, gamma=self.Lognormal_3P_gamma).PDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma , \gamma$)')
        Normal_Distribution(mu=self.Normal_2P_mu, sigma=self.Normal_2P_sigma).PDF(xvals=xvals, label=r'Normal ($\mu , \sigma$)')
        if max(X) <= 1:  # condition for Beta Dist to be fitted
            Beta_Distribution(alpha=self.Beta_2P_alpha, beta=self.Beta_2P_beta).PDF(xvals=xvals, label=r'Beta ($\alpha , \beta$)')
        plt.legend()
        plt.xlim([xmin, xmax])
        plt.title('Probability Density Function')
        plt.xlabel('Data')
        plt.ylabel('Probability density')
        plt.legend()

        plt.subplot(122)  # CDF
        plt.bar(center, hist_cumulative * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')
        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')
        Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta , \gamma$)')

# make this histogram. Can't use plt.hist due to need to scale the heights when there's censored data
        num_bins = min(int(len(X) / 2), 30)
        hist, bins = np.histogram(X, bins=num_bins, density=True)
        hist_cumulative = np.cumsum(hist) / sum(hist)
        width = np.diff(bins)
        center = (bins[:-1] + bins[1:]) / 2
        plt.bar(center, hist * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')

        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).PDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).PDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).PDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')
        Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma).PDF(xvals=xvals, label=r'Gamma ($\alpha , \beta , \gamma$)')
        Exponential_Distribution(Lambda=self.Expon_1P_lambda).PDF(xvals=xvals, label=r'Exponential ($\lambda$)')
        Exponential_Distribution(Lambda=self.Expon_2P_lambda, gamma=self.Expon_2P_gamma).PDF(xvals=xvals, label=r'Exponential ($\lambda , \gamma$)')
        Lognormal_Distribution(mu=self.Lognormal_2P_mu, sigma=self.Lognormal_2P_sigma).PDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma$)')
        Lognormal_Distribution(mu=self.Lognormal_3P_mu, sigma=self.Lognormal_3P_sigma, gamma=self.Lognormal_3P_gamma).PDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma , \gamma$)')
        Normal_Distribution(mu=self.Normal_2P_mu, sigma=self.Normal_2P_sigma).PDF(xvals=xvals, label=r'Normal ($\mu , \sigma$)')
        if max(X) <= 1:  # condition for Beta Dist to be fitted
            Beta_Distribution(alpha=self.Beta_2P_alpha, beta=self.Beta_2P_beta).PDF(xvals=xvals, label=r'Beta ($\alpha , \beta$)')
        plt.legend()
        plt.xlim([xmin, xmax])
        plt.title('Probability Density Function')
        plt.xlabel('Data')
        plt.ylabel('Probability density')
        plt.legend()

        plt.subplot(122)  # CDF
        plt.bar(center, hist_cumulative * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')
        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')

xvals = np.logspace(xmin_log - 2, xmax_log + 2, 1000)

    if __fitted_dist_params is not None:
        if __fitted_dist_params.gamma > 0:
            fit_gamma = True

    if fit_gamma is False:
        if __fitted_dist_params is not None:
            mu = __fitted_dist_params.mu
            sigma = __fitted_dist_params.sigma
        else:
            from reliability.Fitters import Fit_Lognormal_2P
            fit = Fit_Lognormal_2P(failures=failures, right_censored=right_censored, show_probability_plot=False, print_results=False)
            mu = fit.mu
            sigma = fit.sigma
        lnf = Lognormal_Distribution(mu=mu, sigma=sigma).CDF(show_plot=False, xvals=xvals)
        if 'label' in kwargs:
            label = kwargs.pop('label')
        else:
            label = str('Fitted Lognormal_2P (μ=' + str(round_to_decimals(mu, dec)) + ', σ=' + str(round_to_decimals(sigma, dec)) + ')')
        if 'color' in kwargs:
            color = kwargs.pop('color')
            data_color = color
        else:
            color = 'red'
            data_color = 'k'
        plt.xlabel('Time')
    elif fit_gamma is True:
        if __fitted_dist_params is not None:
            mu = __fitted_dist_params.mu
            sigma = __fitted_dist_params.sigma
            gamma = __fitted_dist_params.gamma

elif dist_name == 'Exponential_1P':
                ranked_distributions_objects.append(Exponential_Distribution(Lambda=fitted_results.Expon_1P_lambda))
                ranked_distributions_labels.append(str('Exponential_1P (lambda=' + str(round(fitted_results.Expon_1P_lambda, sigfig)) + ')'))
            elif dist_name == 'Beta_2P':
                ranked_distributions_objects.append(Beta_Distribution(alpha=fitted_results.Beta_2P_alpha, beta=fitted_results.Beta_2P_beta))
                ranked_distributions_labels.append(str('Beta_2P (α=' + str(round(fitted_results.Beta_2P_alpha, sigfig)) + ',β=' + str(round(fitted_results.Beta_2P_beta, sigfig)) + ')'))

            if include_location_shifted is True:
                if dist_name == 'Weibull_3P':
                    ranked_distributions_objects.append(Weibull_Distribution(alpha=fitted_results.Weibull_3P_alpha, beta=fitted_results.Weibull_3P_beta, gamma=fitted_results.Weibull_3P_gamma))
                    ranked_distributions_labels.append(str('Weibull_3P (α=' + str(round(fitted_results.Weibull_3P_alpha, sigfig)) + ',β=' + str(round(fitted_results.Weibull_3P_beta, sigfig)) + ',γ=' + str(round(fitted_results.Weibull_3P_gamma, sigfig)) + ')'))
                elif dist_name == 'Gamma_3P':
                    ranked_distributions_objects.append(Gamma_Distribution(alpha=fitted_results.Gamma_3P_alpha, beta=fitted_results.Gamma_3P_beta, gamma=fitted_results.Gamma_3P_gamma))
                    ranked_distributions_labels.append(str('Gamma_3P (α=' + str(round(fitted_results.Gamma_3P_alpha, sigfig)) + ',β=' + str(round(fitted_results.Gamma_3P_beta, sigfig)) + ',γ=' + str(round(fitted_results.Gamma_3P_gamma, sigfig)) + ')'))
                elif dist_name == 'Lognormal_3P':
                    ranked_distributions_objects.append(Lognormal_Distribution(mu=fitted_results.Lognormal_3P_mu, sigma=fitted_results.Lognormal_3P_sigma, gamma=fitted_results.Lognormal_3P_gamma))
                    ranked_distributions_labels.append(str('Lognormal_3P (μ=' + str(round(fitted_results.Lognormal_3P_mu, sigfig)) + ',σ=' + str(round(fitted_results.Lognormal_3P_sigma, sigfig)) + ',γ=' + str(round(fitted_results.Lognormal_3P_gamma, sigfig)) + ')'))
                elif dist_name == 'Exponential_2P':
                    ranked_distributions_objects.append(Exponential_Distribution(Lambda=fitted_results.Expon_1P_lambda, gamma=fitted_results.Expon_2P_gamma))
                    ranked_distributions_labels.append(str('Exponential_1P (lambda=' + str(round(fitted_results.Expon_1P_lambda, sigfig)) + ',γ=' + str(round(fitted_results.Expon_2P_gamma, sigfig)) + ')'))

        number_of_distributions_fitted = len(ranked_distributions_objects)
        self.results = ranked_distributions_objects
        self.most_similar_distribution = ranked_distributions_objects[0]
        if print_results is True:
            print('The input distribution was:')
            print(distribution.param_title_long)
            if number_of_distributions_fitted < number_of_distributions_to_show:
                number_of_distributions_to_show = number_of_distributions_fitted
            print('\nThe top', number_of_distributions_to_show, 'most similar distributions are:')
            counter = 0
            while counter < number_of_distributions_to_show and counter < number_of_distributions_fitted:

plt.xlim([xmin, xmax])
        plt.title('Probability Density Function')
        plt.xlabel('Data')
        plt.ylabel('Probability density')
        plt.legend()

        plt.subplot(122)  # CDF
        plt.bar(center, hist_cumulative * self._frac_fail, align='center', width=width, alpha=0.2, color='k', edgecolor='k')
        Weibull_Distribution(alpha=self.Weibull_2P_alpha, beta=self.Weibull_2P_beta).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta$)')
        Weibull_Distribution(alpha=self.Weibull_3P_alpha, beta=self.Weibull_3P_beta, gamma=self.Weibull_3P_gamma).CDF(xvals=xvals, label=r'Weibull ($\alpha , \beta , \gamma$)')
        Gamma_Distribution(alpha=self.Gamma_2P_alpha, beta=self.Gamma_2P_beta).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta$)')
        Gamma_Distribution(alpha=self.Gamma_3P_alpha, beta=self.Gamma_3P_beta, gamma=self.Gamma_3P_gamma).CDF(xvals=xvals, label=r'Gamma ($\alpha , \beta , \gamma$)')
        Exponential_Distribution(Lambda=self.Expon_1P_lambda).CDF(xvals=xvals, label=r'Exponential ($\lambda$)')
        Exponential_Distribution(Lambda=self.Expon_2P_lambda, gamma=self.Expon_2P_gamma).CDF(xvals=xvals, label=r'Exponential ($\lambda , \gamma$)')
        Lognormal_Distribution(mu=self.Lognormal_2P_mu, sigma=self.Lognormal_2P_sigma).CDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma$)')
        Lognormal_Distribution(mu=self.Lognormal_3P_mu, sigma=self.Lognormal_3P_sigma, gamma=self.Lognormal_3P_gamma).CDF(xvals=xvals, label=r'Lognormal ($\mu , \sigma , \gamma$)')
        Normal_Distribution(mu=self.Normal_2P_mu, sigma=self.Normal_2P_sigma).CDF(xvals=xvals, label=r'Normal ($\mu , \sigma$)')
        if max(X) <= 1:  # condition for Beta Dist to be fitted
            Beta_Distribution(alpha=self.Beta_2P_alpha, beta=self.Beta_2P_beta).CDF(xvals=xvals, label=r'Beta ($\alpha , \beta$)')
        plt.legend()
        plt.xlim([xmin, xmax])
        plt.title('Cumulative Distribution Function')
        plt.xlabel('Data')
        plt.ylabel('Cumulative probability density')
        plt.suptitle('Histogram plot of each fitted distribution')
        plt.legend()