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

Given the probability distributions for stress and strength, this module will find the probability of failure due to stress-strength interference.
    Failure is defined as when stress>strength.
    Uses the exact formula method which is only valid for two Normal Distributions.

    Inputs:
    stress - a probability distribution from the Distributions module
    strength - a probability distribution from the Distributions module
    show_distribution_plot - True/False (default is True)
    print_results - True/False (default is True)

    Returns:
    the probability of failure
    '''
    if type(stress) is not Normal_Distribution:
        raise ValueError('Both stress and strength must be a Normal_Distribution. If you need another distribution then use Probability_of_failure rather than Probability_of_failure_normdist')
    if type(strength) is not Normal_Distribution:
        raise ValueError('Both stress and strength must be a Normal_Distribution. If you need another distribution then use Probability_of_failure rather than Probability_of_failure_normdist')

    sigma_strength = strength.sigma
    mu_strength = strength.mu
    sigma_stress = stress.sigma
    mu_stress = stress.mu
    prob_of_failure = ss.norm.cdf(-(mu_strength - mu_stress) / ((sigma_strength ** 2 + sigma_stress ** 2) ** 0.5))

    if print_results is True:
        print('Probability of failure:', prob_of_failure)

    if show_distribution_plot is True:
        xmin = stress.b5
        xmax = strength.b95
        xvals = np.linspace(xmin, xmax, 1000)
        stress_PDF = stress.PDF(xvals=xvals, show_plot=False)

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()

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 ($\alpha , \beta , \gamma$)')
        Exponential_Distribution(Lambda=self.Expon_1P_lambda).CDF(xvals=xvals, label=r'Exponential ($\lambda$)')

self.success = False
            print('WARNING: Fitting using Autograd FAILED for Normal_2P. The fit from Scipy was used instead so results may not be accurate.')
            self.mu = sp[0]
            self.sigma = sp[1]

        params = [self.mu, self.sigma]
        k = len(params)
        n = len(all_data)
        LL2 = 2 * Fit_Normal_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 = Normal_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_Normal_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_Normal_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)

fitted_results = Fit_Everything(failures=RVS_filtered, print_results=False, show_probability_plot=False, show_histogram_plot=False, show_PP_plot=False)  # fit all distributions to the filtered samples
        ranked_distributions = list(fitted_results.results.index.values)
        ranked_distributions.remove(distribution.name2)  # removes the fitted version of the original distribution

        ranked_distributions_objects = []
        ranked_distributions_labels = []
        sigfig = 2
        for dist_name in ranked_distributions:
            if dist_name == 'Weibull_2P':
                ranked_distributions_objects.append(Weibull_Distribution(alpha=fitted_results.Weibull_2P_alpha, beta=fitted_results.Weibull_2P_beta))
                ranked_distributions_labels.append(str('Weibull_2P (α=' + str(round(fitted_results.Weibull_2P_alpha, sigfig)) + ',β=' + str(round(fitted_results.Weibull_2P_beta, sigfig)) + ')'))
            elif dist_name == 'Gamma_2P':
                ranked_distributions_objects.append(Gamma_Distribution(alpha=fitted_results.Gamma_2P_alpha, beta=fitted_results.Gamma_2P_beta))
                ranked_distributions_labels.append(str('Gamma_2P (α=' + str(round(fitted_results.Gamma_2P_alpha, sigfig)) + ',β=' + str(round(fitted_results.Gamma_2P_beta, sigfig)) + ')'))
            elif dist_name == 'Normal_2P':
                ranked_distributions_objects.append(Normal_Distribution(mu=fitted_results.Normal_2P_mu, sigma=fitted_results.Normal_2P_sigma))
                ranked_distributions_labels.append(str('Normal_2P (μ=' + str(round(fitted_results.Normal_2P_mu, sigfig)) + ',σ=' + str(round(fitted_results.Normal_2P_sigma, sigfig)) + ')'))
            elif dist_name == 'Lognormal_2P':
                ranked_distributions_objects.append(Lognormal_Distribution(mu=fitted_results.Lognormal_2P_mu, sigma=fitted_results.Lognormal_2P_sigma))
                ranked_distributions_labels.append(str('Lognormal_2P (μ=' + str(round(fitted_results.Lognormal_2P_mu, sigfig)) + ',σ=' + str(round(fitted_results.Lognormal_2P_sigma, sigfig)) + ')'))
            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':

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:
            Fit_Everything.histogram_plot(self)  # plotting occurs by default

        if show_PP_plot is True:
            Fit_Everything.P_P_plot(self)  # plotting occurs by default

        if show_probability_plot is True:

from reliability.Fitters import Fit_Normal_2P
        fit = Fit_Normal_2P(failures=failures, right_censored=right_censored, show_probability_plot=False, print_results=False)
        mu = fit.mu
        sigma = fit.sigma
    if 'label' in kwargs:
        label = kwargs.pop('label')
    else:
        label = str('Fitted Normal_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.scatter(x, y, marker='.', linewidth=2, c=data_color)
    nf = Normal_Distribution(mu=mu, sigma=sigma).CDF(show_plot=False, xvals=xvals)
    xrange = plt.gca().get_xlim()  # this ensures the previously plotted objects are considered when setting the range
    xrange_min = min(min(x) - delta * 0.2, xrange[0])
    xrange_max = max(max(x) + delta * 0.2, xrange[1])
    plt.xlim([xrange_min, xrange_max])
    plt.title('Probability plot\nNormal CDF')
    plt.xlabel('Time')
    plt.ylabel('Fraction failing')
    plt.gcf().set_size_inches(9, 7)  # adjust the figsize. This is done post figure creation so that layering is easier
    if show_fitted_distribution is True:
        plt.plot(xvals, nf, color=color, label=label, **kwargs)
        plt.legend(loc='upper left')
    return plt.gcf()

Note that the empirical CDF also accepts X_data_right_censored just as Kaplan-Meier and Nelson-Aalen will also accept right censored data.

    Inputs:
    X_data_failures - the failure times in an array or list
    X_data_right_censored - the right censored failure times in an array or list. Optional input.
    Y_dist - a probability distribution. The CDF of this distribution will be plotted along the Y-axis.
    method - 'KM' or 'NA' for Kaplan-Meier and Nelson-Aalen. Default is 'KM'
    show_diagonal_line - True/False. Default is True. If True the diagonal line will be shown on the plot.

    Outputs:
    The PP_plot is the only output. Use plt.show() to show it.
    '''

    if X_data_failures is None or Y_dist is None:
        raise ValueError('X_data_failures and Y_dist must both be specified. X_data_failures can be an array or list of failure times. Y_dist must be a probability distribution generated using the Distributions module')
    if type(Y_dist) not in [Weibull_Distribution, Normal_Distribution, Lognormal_Distribution, Exponential_Distribution, Gamma_Distribution, Beta_Distribution] or type(Y_dist) not in [Weibull_Distribution, Normal_Distribution, Lognormal_Distribution, Exponential_Distribution, Gamma_Distribution, Beta_Distribution]:
        raise ValueError('Y_dist must be specified as a probability distribution generated using the Distributions module')
    if type(X_data_failures) == list:
        X_data_failures = np.sort(np.array(X_data_failures))
    elif type(X_data_failures) == np.ndarray:
        X_data_failures = np.sort(X_data_failures)
    else:
        raise ValueError('X_data_failures must be an array or list')
    if type(X_data_right_censored) == list:
        X_data_right_censored = np.sort(np.array(X_data_right_censored))
    elif type(X_data_right_censored) == np.ndarray:
        X_data_right_censored = np.sort(X_data_right_censored)
    elif X_data_right_censored is None:
        pass
    else:
        raise ValueError('X_data_right_censored must be an array or list')
    # extract certain keyword arguments or specify them if they are not set