Understanding Key Statistical Concepts in Simple Terms 📊

What is a Random Variable? 🤔

A random variable is a way to assign numbers to different outcomes of an experiment.

💡 Example: Tossing a coin 🎲

• Heads → 1

• Tails → 0

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Types of Probability Distributions 🎯

Probability distributions tell us how likely different outcomes are. They can be:

1️⃣ Probability Density Function (PDF) – Continuous Data 📈

Used for variables that can take any value in a range.

• Normal Distribution (Bell Curve) 🔔

• Standard Normal Distribution

• Log-Normal Distribution

• Chi-Square & F-Distribution

2️⃣ Probability Mass Function (PMF) – Discrete Data 🎲

Used for variables that take specific values.

• Bernoulli Distribution ✅❌ (Yes/No)

• Binomial Distribution 🔢 (Multiple Yes/No)

• Poisson Distribution 📞 (Rare events happening over time)

3️⃣ Uniform Distribution (Equal Chance for All) 🔄

• Discrete Uniform Distribution (Rolling a fair die 🎲)

• Continuous Uniform Distribution (Random number generator 🔢)

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PMF vs. PDF vs. CDF 🤷‍♂️

1️⃣ PMF (Probability Mass Function) 🎲

• Used for discrete random variables (like rolling a die)

• Example: Each face of a fair die has a probability of 1/6.

• Graph:

  

2️⃣ PDF (Probability Density Function) 📈

• Used for continuous variables.

• Example: Height of people → Bell Curve (Normal Distribution)

• Graph:

  

3️⃣ CDF (Cumulative Distribution Function) 📊

• Tells the probability that a value is less than or equal to x.

• Example: The probability of rolling ≤ 3 on a die.

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Understanding Different Distributions 📚

🎲 Uniform Distribution

• Discrete Example: Rolling a fair die → Each number is equally likely.

• Continuous Example: A random number generator selecting values between 0 and 1.

• Question: If a shop sells between 20-50 items daily, what is the probability of selling between 25 and 40?

✅ Bernoulli Distribution

• Definition: Used when there are only two possible outcomes.

• Example: Flipping a coin (Heads/Tails).

• Mean = p, Variance = p(1-p)

📞 Poisson Distribution

• Definition: Used when counting rare events over a fixed time.

• Example: Number of calls received per hour in a call center.

• Why It Matters: Poisson is useful when counting rare events that happen at a constant rate over time, like the number of customer complaints received daily in a store.

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Normal Distribution (Bell Curve) 🔔

Characteristics:

1️⃣ Symmetric around the mean.

2️⃣ Mean = Median = Mode.

3️⃣ No skewness.

Empirical Rule (68-95-99.7 Rule) 📊

• 68% of values fall within 1 standard deviation.

• 95% within 2 standard deviations.

• 99.7% within 3 standard deviations.

Standard Normal Distribution (SND) ❓

• Why use SND when we have Normal Distribution?

• SND is a special case of Normal Distribution where mean = 0 and standard deviation = 1.

• Use Case: Helps standardize data for comparisons and is essential for ML models like Linear & Logistic Regression.

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Central Limit Theorem (CLT) 🧠

What is CLT?

• If we take many samples from a population, the average of those samples will form a normal distribution.

• Why important? Helps in making predictions & confidence intervals.

• Graphical Representation:

🔹 Conditions for CLT:

1️⃣ The sample size should be large.

2️⃣ Sample size ≥ 30.

Real-World Application:

• CLT is widely used in A/B testing, where companies test different versions of a website and analyze user engagement. Since individual user behaviors can vary, taking many samples ensures a normal distribution of results.

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Understanding Standard Error 🤓

• What is it? It tells us how much the sample mean differs from the population mean.

• Why important? Used in confidence intervals & hypothesis testing.

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Z-Score Applications 📏

• What is a Z-score? It measures how far a value is from the mean.

• Example Question: Given marks X = {1,2,3,4,5,6}, mean = 4, SD = 1, find probability that score > 4.5.

Real-World Application:

• Credit Scoring: Banks use Z-scores to determine how risky a loan applicant is. If an applicant's credit score is far below the mean, they may be considered high-risk.

• Medical Diagnosis: Z-scores are used in bone density tests to identify osteoporosis. A low Z-score indicates weaker bones compared to the average population.

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Point Estimate vs. Interval Estimate 🎯

Point Estimate

• Example: Finding the average salary of IT employees.

Interval Estimate

• Gives a range instead of a single number.

• Example: Predicting an IT employee’s salary will be between $50K and $70K.

🔹 When to use what?

• Use Point Estimate when a single number is needed.

• Use Interval Estimate when uncertainty is involved.

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Final Thoughts 🚀

These statistical concepts are the foundation of data analysis and machine learning. Understanding them will help in making data-driven decisions! 🎯