Probability Distributions

2. Overview of Common Distributions

2.1 Common Distributions
1. An overview of the most common probability distributions:

2.2 Quick Example
1. Suppose we run a cake shop: ➜ Daily sales amount may be modeled using a Normal distribution. ➜ Whether one visitor buys a cake is a Bernoulli outcome. ➜ Number of buyers among 100 visitors may follow a Binomial. ➜ Number of customer arrivals per hour may follow Poisson. 📌 The business event determines which distribution may be appropriate.

3. Normal Distribution

3.1 🔔 What is a Normal Distribution?
1. The Normal distribution is a continuous probability distribution with a symmetric, bell-shaped curve. It is described by two parameters: mean (μ), standard deviation (σ). - The mean determines the center of the distribution. - The standard deviation determines how widely values are spread around the mean.
3.2 Main Characteristics
1. A Normal distribution is: ✅ Bell-shaped ✅ Symmetric around the mean ✅ Centered at μ ✅ Controlled in spread by σ ✅ Continuous. In a perfectly Normal distribution: Mean = Median = Mode
3.3 Effect of Mean and Standard Deviation
1. ➜ Changing the mean moves the curve to the left or right. ➜ Changing the standard deviation changes the spread: - Smaller standard deviation produces a narrower curve - Larger standard deviation produces a wider curve. 📌 The mean controls location. The standard deviation controls spread.
3.4 The 68–95–99.7 Rule
1. For a Normal distribution, approximately: - 68% of values lie within 1 standard deviation of the mean - 95% lie within 2 standard deviations - 99.7% lie within 3 standard deviations. We can write this as: μ ± 1σ → Approximately 68%μ ± 2σ → Approximately 95%μ ± 3σ → Approximately 99.7%
3.5 Example
1. Suppose student heights have: Mean = 165 cmStandard deviation = 5 cm Approximately 68% of heights would lie between: 165 - 5 = 160 cm165 + 5 = 170 cm Therefore: Approximately 68% lie between 160 cm and 170 cm. 📌 The 68–95–99.7 rule applies when the distribution is approximately Normal.
3.6 Python Codes
1. Download heart_df_clean data. Load it as pandas dataframe. If a person is selected on random then using the dataset answer the questions below: What is probability that he has cholestrol below 250?import pandas as pdimport scipy.stats as statsheart_df_clean = pd.read_csv("heart_df_clean.csv")mean = heart_df_clean["totChol"].mean()std_dev = heart_df_clean["totChol"].std()p_lt_250 = stats.norm.cdf(250, loc=mean, scale=std_dev)print(f"P(X <= 250): {round(p_lt_250 * 100, 2)}%")What is probability that he has cholestrol above 280?p_gt_280 = 1 - stats.norm.cdf(280, loc=mean, scale=std_dev)print(f"P(X > 280): {round(p_gt_280 * 100, 2)}%")What is probability that he has cholestrol between 230 and 250?p_le_250 = stats.norm.cdf(250, loc=mean, scale=std_dev)p_le_230 = stats.norm.cdf(230, loc=mean, scale=std_dev)p_btn_230_250 = p_le_250 - p_le_230print(f"P(230 <= X <= 250): {round(p_btn_230_250 * 100, 2)}%")

4. Bernoulli Distribution

4.1 🪙 What is a Bernoulli Distribution?
1. A Bernoulli distribution models one trial with two possible outcomes: Success = 1Failure = 0 It has one main parameter: p = Probability of success The probability of failure is: 1 − p
4.2 Bernoulli Examples
1. Coin Toss: Success: Head Failure: Tail p = 0.5Customer Purchase: Success: Customer purchases Failure: Customer does not purchase p = 0.2Email Open: Success: Email is opened Failure: Email is not opened p = 0.3 📌 A Bernoulli distribution describes one yes/no or success/failure trial.
4.3 Possible Values
1. A Bernoulli random variable can take only two values: X = 1 with probability pX = 0 with probability 1 - pExample:P(X = 1) = 0.3P(X = 0) = 0.7
4.4 Python Code
1. Download heart_df_clean data. Load it as pandas dataframe. If a person is selected on random then using the dataset answer the questions below: What is probability that he has heart disease?import pandas as pdheart_df_clean = pd.read_csv("heart_df_clean.csv")p = heart_df_clean["target"].mean()print(f"P(X = 1): {round(p * 100, 2)}%")What is probability that he does not have heart disease?probability_no_disease = 1 - pprint(f"P(X = 0): {round(probability_no_disease * 100, 2)}%")

5. Binomial Distribution

5.1 🎯 What is a Binomial Distribution?
1. A Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. Its two main parameters are: n = Number of trialsp = Probability of success in each trial A Binomial random variable may take values from: 0 to n
5.2 Conditions for a Binomial Distribution
1. A Binomial model generally requires: ✅ A fixed number of trials ✅ Two possible outcomes per trial ✅ The same success probability in every trial ✅ Independent trials.
5.3 Example: Coin Tosses
1. Suppose we toss a fair coin 10 times. n = 10p = 0.5 The random variable may be: X = Number of heads in 10 tosses Possible values are: 0, 1, 2, ..., 10
5.4 Business Example
1. Suppose 100 people visit a website, and each visitor has a 20% probability of purchasing. n = 100p = 0.2 The random variable is: X = Number of visitors who purchase 📌 Bernoulli models one purchase decision. 📌 Binomial models the total number of purchases across multiple trials.
5.5 Expected Number of Successes
1. The mean or expected value of a Binomial distribution is: Mean = n × p. For, n = 100p = 0.2 The expected number of purchases is: 100 × 0.2 = 20 This does not mean exactly 20 customers will always purchase. It is the long-run average expected under the model.
5.6 Python Code
1. Download heart_df_clean data. Load it as pandas dataframe. If a person is selected on random then using the dataset answer the questions below: What is probability that exactly 3 people have heart disease in a sample of 10?import pandas as pdfrom scipy.stats import binomheart_df_clean = pd.read_csv("heart_df_clean.csv")p = heart_df_clean["target"].mean()n = 10p_eq_3 = binom.pmf(3, n, p)print(f"P(X = 3): {round(p_eq_3 * 100, 2)}%")What is probability that at most 2 people have heart disease in a sample of 10?p_at_most_2 = binom.cdf(2, n, p)print(f"P(X <= 2): {round(p_at_most_2 * 100, 2)}%")What is probability that at least 4 people have heart disease in a sample of 10?p_at_least_4 = 1 - binom.cdf(3, n, p)print(f"P(X >= 4): {round(p_at_least_4 * 100, 2)}%")What is probability that 3 to 5 people have heart disease in a sample of 10?p_3_to_5 = binom.cdf(5, n, p) - binom.cdf(2, n, p)print(f"P(3 <= X <= 5): {round(p_3_to_5 * 100, 2)}%") 📌 cdf of 3 includes probability for 3 which we don't want to delete. Thus, we subtracted cdf of 2 instead.

6. Poisson Distribution

6.1 📞 What is a Poisson Distribution?
1. A Poisson distribution models the number of events occurring within a fixed interval of: time, space, area, distance. Its main parameter is: λ = Average number of events per interval A Poisson model generally assumes: ✅ Events occur independently ✅ The average rate (λ) is constant ✅ Events are counted within a fixed interval
6.2 Poisson Examples
1. Examples include: ➜ Number of customers arriving per hour ➜ Number of emails received per day ➜ Number of calls received per minute ➜ Number of defects per meter of material ➜ Number of accidents at a location per month
6.3 Example
1. Suppose a cake shop receives an average of 12 customers per hour. λ = 12 customers per hour A Poisson model could be used to calculate the probability of receiving: ➜ Exactly 10 customers ➜ Fewer than 8 customers ➜ More than 15 customers. 📌 The value of λ must use the same interval as the question.
6.4 Expected Number of Events
1. For a Poisson distribution: Mean = λ If λ = 12, then the expected number of events in that interval is: 12
6.5 Python Code
1. Suppose 20 people visit a website per hour on average. Calculate probabilities: What is probability that exactly 15 people visit in an hour?import pandas as pdfrom scipy.stats import poissonlambda_value = 20p_eq_15 = poisson.pmf(15, lambda_value)print(f"P(X = 15): {round(p_eq_15 * 100, 2)}%")What is probability that at most 18 people visit in an hour?p_le_18 = poisson.cdf(18, lambda_value)print(f"P(X <= 18): {round(p_le_18 * 100, 2)}%")What is probability that at least 25 people visit in an hour?p_ge_25 = 1 - poisson.cdf(24, lambda_value)print(f"P(X >= 25): {round(p_ge_25 * 100, 2)}%")What is probability that between 15 and 25 people visit in an hour?p_le_14 = poisson.cdf(14, lambda_value)p_le_25 = poisson.cdf(25, lambda_value)p_btn_15_25 = p_le_25 - p_le_14print(f"P(15 <= X <= 25): {round(p_btn_15_25 * 100, 2)}%")

7. Estimating Distribution Parameters

7.1 📏 Where Do Parameters Come From?
1. Some parameters are estimated from historical or observed data, while others are fixed by the experiment. Common parameters include: ➜ p is estimated for Bernoulli and Binomial distributions ➜ n is the fixed or known number of Binomial trials ➜ λ is estimated for the Poisson distribution ➜ μ and σ are estimated for the Normal distribution
7.2 Estimating p
1. We estimate the probability of success using: p = Number of successes / Total number of trialsExample: Purchases = 40 Visitors = 200Therefore: p = 40 / 200 p = 0.2 The estimated purchase probability is: 20%
7.3 Estimating λ
1. We estimate λ using the average number of events per interval. Example customer arrivals: Hour 1: 10Hour 2: 14Hour 3: 12Hour 4: 8Hour 5: 16 Here, λ = (10 + 14 + 12 + 8 + 16) / 5λ = 12 The estimated average arrival rate is: 12 customers per hour
7.4 Estimating μ and σ
1. For a Normal model: - Estimate μ using the sample mean - Estimate σ using the sample standard deviation. 📌 Parameters summarize important features of a probability distribution.

8. Choosing the Appropriate Distribution

8.1 🍰 Cake Shop Example
1. A distribution selection guide for a cake shop:

8.2 Distribution Selection Questions
1. Ask the following questions: Q: Is the variable continuous and approximately symmetric? A: Consider: Normal distribution Q: Is there one trial with two outcomes? A: Consider: Bernoulli distribution Q: Are we counting successes across a fixed number of trials? A: Consider: Binomial distribution Q: Are we counting independent events occurring at an approximately constant rate within a fixed interval? A: Consider: Poisson distribution 📌 Start by identifying the type of outcome and how the data is generated.

9. PMF, PDF, and CDF

9.1 📊 Why Do We Need These Functions?
1. Probability distributions use different functions to describe probabilities. The main functions are: - Probability Mass Function (PMF) - Probability Density Function (PDF) - Cumulative Distribution Function (CDF).
9.2 Probability Mass Function
1. The Probability Mass Function, or PMF, gives the probability of each possible value of a discrete random variable. Applicable to: ✅ Bernoulli distribution ✅ Binomial distribution ✅ Poisson distribution 📌 For discrete distributions, an individual value can have a nonzero probability.
9.3 Probability Density Function
1. The Probability Density Function, or PDF, describes probability density for a continuous random variable. Applicable to: ✅ Normal distribution ✅ Exponential distribution For a continuous variable: P(X = exactly one value) = 0 Probability is calculated over an interval: P(a ≤ X ≤ b) This probability corresponds to the area under the density curve between a and b. 📌 PDF height represents density. Probability comes from an area under the curve.
9.4 Cumulative Distribution Function
1. The Cumulative Distribution Function, or CDF, gives: P(X ≤ x) It is the probability that a random variable is less than or equal to a specified value. The CDF applies to both: ✅ Discrete distributions ✅ Continuous distributions.

10. Confidence Intervals

10.1 🎯 What is a Confidence Interval?
1. A confidence interval is a range calculated from sample data and used to estimate an unknown population parameter. Examples of population parameters include: ➜ Population mean ➜ Population proportion ➜ Difference between two population means A point estimate gives one estimated value. A confidence interval provides a range together with an indication of uncertainty.
10.2 Point Estimate vs. Interval Estimate
1. Suppose we want to estimate the average number of fish in a lake. Point Estimate: A point estimate provides one value: Estimated average = 500 fishConfidence Interval: A confidence interval provides a range: Estimated range = 450 to 550 fish 📌 A point estimate gives one best estimate. 📌 A confidence interval shows the uncertainty around that estimate.
10.3 Net Analogy
1. Imagine the unknown population mean is a fish in a lake. - A point estimate is like throwing one spear toward the fish. - A confidence interval is like casting a net around the estimated location. A method that produces wider intervals generally has greater coverage but gives less precise estimates. A method that produces narrower intervals gives more precise estimates but may have lower coverage.
10.4 General Confidence Interval Formula
1. The general structure is: Confidence Interval = Point Estimate ± Margin of Error The margin of error is: Margin of Error = Critical Value × Standard Error Therefore: CI = Point Estimate ± Critical Value × Standard Error
10.5 Confidence Interval for a Mean
1. When using a Normal critical value and a known population standard deviation: CI = X̄ ± Z × SE where: X̄ = Sample meanZ = Critical valueSE = Standard error The standard error is: SE = σ / √n where: σ = Population standard deviationn = Sample size For a 95% confidence level: Z ≈ 1.96 A Normal-based confidence interval for a population mean with known population standard deviation uses this structure.
10.6 Python Code
1. Download heart_df_clean data. Load it as pandas dataframe and answer the questions below: Calculate 95% confidence interval for mean BMIimport pandas as pdimport scipy.stats as statsheart_df_clean = pd.read_csv("heart_df_clean.csv")mean_bmi = heart_df_clean["BMI"].mean()std_dev_bmi = heart_df_clean["BMI"].std()n = len(heart_df_clean)std_error = stats.sem(heart_df_clean["BMI"])ci = stats.norm.interval(0.95, loc=mean_bmi, scale=std_error)lower_bound = ci[0]upper_bound = ci[1]print(f"95% Confidence Interval for Mean BMI: ({lower_bound:.2f}, {upper_bound:.2f})")
10.7 Interpreting a 95% Confidence Level
1. A 95% confidence level describes the long-run performance of the method. If we repeatedly collected samples and constructed intervals using the same method, approximately 95% of those intervals would contain the true population parameter. It does not mean that a particular calculated interval has a 95% probability of containing the fixed parameter. 📌 Confidence refers to the interval-building process, not to changing probability for a fixed population parameter.
10.8 Important Note about Z and t
1. The formula using Z = 1.96 is appropriate for a Normal-based confidence interval when the population standard deviation is known. In many real situations, the population standard deviation is unknown. A Student's t-distribution may then be used, especially for smaller samples. For this introductory lesson: Known population standard deviation → Z-based interval Unknown population standard deviation → Often t-based interval

11. Summary