Probability density and cumulative functions in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the normal distribution from scipy.stats.

SciPy

from scipy.stats import [1]

Drag options to blanks, or click blank then click option'

Astats

Bnormal

Cnorm

Ddistribution

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'normal' instead of 'norm' causes import errors.

Importing 'stats' or 'distribution' is incorrect for normal distribution.

The normal distribution in scipy.stats is imported as norm.

2fill in blank

medium

Complete the code to calculate the probability density function (PDF) at x=0 for a standard normal distribution.

SciPy

pdf_value = norm.[1](0)

Drag options to blanks, or click blank then click option'

Appf

Bcdf

Cpmf

Dpdf

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'cdf' instead of 'pdf' returns cumulative probability, not density.

Using 'pmf' is for discrete distributions, not normal distribution.

The pdf method calculates the probability density function value at a given point.

3fill in blank

hard

Fix the error in the code to compute the cumulative distribution function (CDF) at x=1.5.

SciPy

cdf_value = norm.[1](1.5)

Drag options to blanks, or click blank then click option'

Apdf

Bcdf

Cppf

Drvs

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'pdf' returns density, not cumulative probability.

Using 'ppf' returns quantiles, not cumulative probability.

The cdf method returns the cumulative probability up to the given value.

4fill in blank

hard

Fill both blanks to create a dictionary of PDF values for x in [0, 1, 2] where PDF is greater than 0.1.

SciPy

pdf_dict = {x: norm.[1](x) for x in [0, 1, 2] if norm.[2](x) > 0.1}

Drag options to blanks, or click blank then click option'

Apdf

Bcdf

Crvs

Dppf

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'cdf' in the filter returns cumulative probabilities, not densities.

Using 'rvs' or 'ppf' are unrelated to density calculation.

We use pdf to get the density values and filter those greater than 0.1.

5fill in blank

hard

Fill all three blanks to create a dictionary of cumulative probabilities for x in [0, 1, 2] where the cumulative probability is less than 0.9.

SciPy

cdf_dict = {x[1]2: norm.[2](x) for x in [0, 1, 2] if norm.[3](x) < 0.9}

Drag options to blanks, or click blank then click option'

A*

Bcdf

C<

D**

Attempts:

3 left

💡 Hint

Common Mistakes

Using multiplication (*) instead of exponentiation (**) for keys.

Using 'pdf' instead of 'cdf' for cumulative probabilities.

The dictionary keys are x squared (**), values are cumulative probabilities (cdf), filtered where cumulative probability is less than 0.9.

Practice

1. What does the Probability Density Function (PDF) represent in a data distribution?

easy

A. The average value of the data

B. The likelihood of a value occurring at a specific point

C. The total probability up to a certain value

D. The maximum value in the data

Solution

1. Step 1: Understand PDF meaning

   The PDF shows how dense or likely values are at specific points in a distribution.

2. Step 2: Differentiate from CDF

   The CDF shows total probability up to a point, not the likelihood at a point.

3. Final Answer:

   The likelihood of a value occurring at a specific point -> Option B

4. Quick Check:

   PDF = Likelihood at point [OK]

Hint: PDF shows likelihood at exact points, not cumulative [OK]

Common Mistakes:

• Confusing PDF with CDF
• Thinking PDF gives total probability
• Assuming PDF gives average or max value

2. Which of the following is the correct way to import the cumulative distribution function (CDF) for a normal distribution from scipy.stats?

easy

A. import norm from scipy.stats; norm.cdf(x)

B. import scipy.stats.norm as norm; norm.cdf(x)

C. from scipy import norm; norm.cdf(x)

D. from scipy.stats import norm; norm.cdf(x)

Solution

1. Step 1: Recall scipy.stats import syntax

   The correct import for normal distribution functions is from scipy.stats import norm.

2. Step 2: Use norm.cdf(x) to get CDF

   After import, norm.cdf(x) computes the cumulative distribution function at x.

3. Final Answer:

   from scipy.stats import norm; norm.cdf(x) -> Option D

4. Quick Check:

   Correct import and usage = from scipy.stats import norm; norm.cdf(x) [OK]

Hint: Use 'from scipy.stats import norm' for normal distribution functions [OK]

Common Mistakes:

• Wrong import syntax causing errors
• Using 'import scipy.stats.norm as norm' which is invalid
• Trying to import norm directly from scipy

3. What is the output of the following code snippet?

from scipy.stats import norm
result = norm.pdf(0, loc=0, scale=1)
print(round(result, 3))

medium

A. 0.399

B. 0.500

C. 1.000

D. 0.000

Solution

1. Step 1: Understand norm.pdf at mean

   For a normal distribution with mean 0 and std 1, PDF at 0 is 1/sqrt(2*pi) ≈ 0.3989.

2. Step 2: Round the result

   Rounding 0.3989 to 3 decimals gives 0.399.

3. Final Answer:

   0.399 -> Option A

4. Quick Check:

   PDF at mean = 0.399 [OK]

Hint: PDF at mean of standard normal ≈ 0.399 [OK]

Common Mistakes:

• Confusing PDF with CDF value
• Rounding incorrectly
• Assuming PDF at mean is 0.5

4. Identify the error in this code that tries to calculate the CDF of a normal distribution at x=1:

import scipy.stats as stats
cdf_value = stats.norm.pdf(1, loc=0, scale=1)
print(cdf_value)

medium

A. Using pdf() instead of cdf() to get cumulative probability

B. Missing import statement for norm

C. Incorrect parameters for norm.pdf

D. Syntax error in print statement

Solution

1. Step 1: Check function used for CDF calculation

   The code uses norm.pdf which gives probability density, not cumulative probability.

2. Step 2: Correct function for cumulative probability

   To get cumulative probability, norm.cdf should be used instead of norm.pdf.

3. Final Answer:

   Using pdf() instead of cdf() to get cumulative probability -> Option A

4. Quick Check:

   Use cdf() for cumulative probability [OK]

Hint: Use cdf() for cumulative, pdf() for density [OK]

Common Mistakes:

• Mixing pdf and cdf functions
• Assuming pdf returns cumulative values
• Ignoring function purpose

5. You have a dataset of exam scores that roughly follow a normal distribution with mean 70 and standard deviation 10. Using scipy, how would you find the probability that a student scored less than 85?

hard

A. Use norm.cdf(70, loc=85, scale=10) to get cumulative probability

B. Use norm.pdf(85, loc=70, scale=10) to get cumulative probability

C. Use norm.cdf(85, loc=70, scale=10) to get cumulative probability

D. Use norm.pdf(70, loc=85, scale=10) to get cumulative probability

Solution

1. Step 1: Identify correct function for cumulative probability

   To find probability of scoring less than 85, use the cumulative distribution function norm.cdf.

2. Step 2: Use correct parameters for mean and std

   Set loc=70 (mean) and scale=10 (std deviation), and evaluate at 85.

3. Final Answer:

   Use norm.cdf(85, loc=70, scale=10) to get cumulative probability -> Option C

4. Quick Check:

   norm.cdf(value, loc=mean, scale=std) = cumulative probability [OK]

Hint: CDF with loc=mean, scale=std, value=threshold [OK]

Common Mistakes:

• Using pdf instead of cdf for cumulative probability
• Swapping mean and value parameters
• Confusing scale with loc