Probability density and cumulative functions in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a Probability Density Function (PDF)?

A PDF shows how likely it is to find a value near a specific point in a continuous distribution. The area under the PDF curve between two points gives the probability of the value falling in that range.

beginner

What does the Cumulative Distribution Function (CDF) represent?

The CDF gives the probability that a random variable is less than or equal to a certain value. It is the area under the PDF curve from the smallest value up to that point.

intermediate

How do you calculate the PDF and CDF using scipy for a normal distribution?

Use scipy.stats.norm. For PDF: norm.pdf(x, loc=mean, scale=std). For CDF: norm.cdf(x, loc=mean, scale=std). Replace x with the value, mean and std with distribution parameters.

beginner

Why is the area under the PDF curve always equal to 1?

Because the total probability of all possible outcomes must be 1. The PDF curve represents how this total probability is spread over values.

intermediate

What is the relationship between PDF and CDF?

The CDF is the integral (sum) of the PDF from the lowest value up to a point. The PDF is the derivative (rate of change) of the CDF.

What does the CDF value at x represent?

A Probability that the variable is exactly x

B Mean of the distribution

C Probability density at x

D Probability that the variable is less than or equal to x

The CDF gives the probability that the variable is less than or equal to x.

Which scipy function calculates the PDF of a normal distribution?

A norm.cdf()

B norm.mean()

C norm.pdf()

D norm.var()

norm.pdf() calculates the probability density function values.

What is the total area under a PDF curve?

A 0

B 1

C Depends on the distribution

D Infinity

The total area under the PDF curve is always 1, representing total probability.

If PDF is the derivative of CDF, what is CDF in terms of PDF?

A Integral of PDF

B Derivative of PDF

C Square of PDF

D Logarithm of PDF

CDF is the integral (area under curve) of the PDF.

Which of these is true for a continuous random variable?

A Probability at a single point is zero

B Probability at a single point is one

C PDF values are probabilities

D CDF decreases as x increases

For continuous variables, probability at an exact point is zero; probabilities come from intervals.

Explain the difference between Probability Density Function (PDF) and Cumulative Distribution Function (CDF).

Think about how probability is spread and accumulated.

Describe how to use scipy to find PDF and CDF values for a normal distribution at a given point.

Recall the function names and parameters.

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