Probability density and cumulative functions in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of PDF calculation for normal distribution

What is the output of this code snippet that calculates the probability density function (PDF) of a normal distribution at x=0?

SciPy

from scipy.stats import norm
result = norm.pdf(0, loc=0, scale=1)
print(round(result, 4))

A 0.3989

B 0.5000

C 1.0000

D 0.0000

💡 Hint

Recall the formula for the PDF of a standard normal distribution at zero.

The PDF of a standard normal distribution at 0 is 1 divided by the square root of 2π, approximately 0.3989.

❓ data_output

intermediate

Cumulative distribution function values for uniform distribution

What is the output array of cumulative distribution function (CDF) values for a uniform distribution between 0 and 10 at points [0, 5, 10, 15]?

SciPy

from scipy.stats import uniform
import numpy as np
points = np.array([0, 5, 10, 15])
cdf_values = uniform.cdf(points, loc=0, scale=10)
print(np.round(cdf_values, 2))

A [0.00 0.25 0.50 0.75]

B [0.00 0.40 0.80 1.00]

C [0.00 0.50 1.00 1.00]

D [0.10 0.50 0.90 1.00]

💡 Hint

CDF of uniform distribution increases linearly from 0 to 1 over the scale.

The uniform CDF at 0 is 0, at 5 is 0.5, at 10 is 1, and beyond 10 remains 1.

❓ visualization

advanced

Plot PDF and CDF of exponential distribution

Which option produces the correct plot showing both the PDF and CDF of an exponential distribution with rate 1 over the range 0 to 5?

SciPy

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import expon
x = np.linspace(0, 5, 100)
plt.plot(x, expon.pdf(x), label='PDF')
plt.plot(x, expon.cdf(x), label='CDF')
plt.legend()
plt.show()

A Plot with PDF and CDF both starting at 0 and increasing to 1

B Plot with PDF curve starting at 1 and decreasing, CDF curve starting at 0 and increasing to 1

C Plot with PDF curve increasing, CDF curve decreasing

D Plot with flat PDF and CDF curves

💡 Hint

Remember the shape of exponential PDF and CDF functions.

The exponential PDF starts at 1 and decreases exponentially; the CDF starts at 0 and increases to 1.

🧠 Conceptual

advanced

Understanding the relationship between PDF and CDF

Which statement correctly describes the relationship between the probability density function (PDF) and the cumulative distribution function (CDF) of a continuous random variable?

A The PDF is the square of the CDF.

B The PDF is the integral of the CDF from negative infinity to x.

C The PDF and CDF are always equal for all x.

D The CDF is the integral of the PDF from negative infinity to x.

💡 Hint

Think about how cumulative probability accumulates from the density.

The CDF at x is the total probability up to x, which is the integral of the PDF up to x.

🔧 Debug

expert

Identify the error in this code calculating normal distribution CDF

What error does this code produce when trying to calculate the CDF of a normal distribution at x=1?

SciPy

from scipy.stats import norm
result = norm.cdf(1, mean=0, std=1)
print(result)

A TypeError: cdf() got an unexpected keyword argument 'mean'

B ValueError: scale must be positive

C SyntaxError: invalid syntax

D Output: 0.8413

💡 Hint

Check the parameter names for scipy.stats.norm.cdf.

The correct parameter names are 'loc' for mean and 'scale' for standard deviation, not 'mean' or 'std'.