Data Science & Machine Learnin… on Telegram: “This tells us the probability that the score is…”

This tells us the probability that the score is 80 or less.

🔹 8. PMF vs PDF vs CDF

PMF: Used for Discrete data. Represents Probability of an exact outcome

PDF: Used for Continuous data. Represents Probability density

CDF: Used for Discrete & continuous. Represents Probability up to a value

A simple way to remember:

PMF → Exact probability for discrete outcomes

PDF → Density across continuous values

CDF → Cumulative probability up to a value

🔹 9. Example: Discrete Distribution

Suppose a machine produces defective products.

Let: X = Number of defective products

Possible values: 0, 1, 2, 3

Suppose:

P(X=0) = 0.50

P(X=1) = 0.30

P(X=2) = 0.15

P(X=3) = 0.05

Check: 0.50 + 0.30 + 0.15 + 0.05 = 1.00

Therefore, this is a valid probability distribution.

🔹 10. Example: Continuous Distribution

Suppose: X = Customer waiting time

Waiting time could be: 2.1 minutes, 2.15 minutes, 2.157 minutes, 2.1578 minutes...

Because there are infinitely many possible values, we treat it as a continuous random variable.

A PDF can describe how densely the waiting times are distributed.

🔹 11. Normal Distribution ⭐

One of the most important probability distributions in Data Science is the Normal Distribution.

It is often called the bell curve because of its shape.

A normal distribution is characterized by: Mean, Standard deviation

Many natural and measurement-related variables can be approximately normally distributed under suitable conditions.

Examples: Measurement errors, Certain biological measurements, Standardized test scores

🔹 12. Properties of Normal Distribution

For a perfectly symmetric normal distribution: Mean = Median = Mode

The distribution is symmetric around its mean.

A common rule of thumb is the 68–95–99.7 rule:

Within 1 Standard Deviation: Approximately 68%

Within 2 Standard Deviations: Approximately 95%

Within 3 Standard Deviations: Approximately 99.7%

🔹 13. Binomial Distribution

The Binomial Distribution is a discrete probability distribution used when:

There are a fixed number of trials, Each trial has two possible outcomes, The probability of success is constant, Trials are independent.

Examples: Number of successful predictions, Number of heads in coin tosses, Number of defective products in a fixed sample

Example: 10 coin tosses. X = Number of Heads. Possible values: 0, 1, 2, ..., 10

🔹 14. Poisson Distribution

The Poisson Distribution is commonly used to model the number of events occurring within a fixed interval when events occur at a certain average rate under appropriate assumptions.

Examples: Number of customer calls per hour, Number of website visits per minute, Number of machine failures per month, Number of support tickets per day

🔹 15. Why Probability Distributions Matter in Data Science?

Probability distributions help Data Scientists:

✅ Understand data patterns

✅ Detect unusual observations

✅ Model uncertainty

✅ Perform statistical tests

✅ Build predictive models

✅ Simulate data

✅ Estimate probabilities

🔹 16. Python Example

import numpy as np

data = np.random.normal(
    loc=50,
    scale=10,
    size=1000
)

print(data[:5])