Gumbel Distribution Exercise#

Imagine you are concerned with the concentration of an airborne contaminant, $X$, measured in ppm. A previous benchmark study estimates that there is a 10% and 1% chance, respectively, of exceeding 4 and 10 ppm, respectively. You have been asked by the regulatory agency to estimate the contaminant concentration with 0.1% probability of being exceeded. Prior studies suggest that a Gumbel distribution can be used to model contaminant concentration, given by the CDF:

Using the cell blocks below as a guide (and also to check your analysis): Task 1) find the parameters of a Gumbel distribution that matches the information provided above, then, Tasks 2-3) use it to estimate the concentration with exceedance probability 0.1%.

To complete this assignment, you can use numpy, matplotlib and from the math library, log and e (these are imported for you when you inialize the notebook).

from example_gumbel import check_example

Task 0: fill in the appropriate fitting points:

x_1 = _

Task 1: derive the distribution parameters:

    """Compute Gumbel distribution parameters from two points of the CDF.

The cells below will print your parameter values and create a plot to help confirm you have the right implementation.

    plt.plot(x_axis, np.vectorize(gumbel_distribution)(x_axis), linewidth=2)

gumbel_distribution = lambda x: 1 - e**(-e**(-(x - mu)/beta))

Task 2: write a function to solve for the random variable value (it will be tested for you with the Check answer button, along with the distribution parameters).

    """ Compute point in the gumbel distribution for which the CDF is p

Task 3: use the function find_x_with_probability_p to evaluate the random variable value with exceedance probabiltiy of 0.001.

print(f"The value of x with probability of exceedance 0.001 os {x:0.5f}")

check_example(globals())