# Study Notes: Discrete Probability Distribution ## Key Concepts ### What is a Discrete Probability Distribution? A **discrete probability distribution** represents the probabilities of outcomes in situations where values are distinct and countable. It is used to analyze datasets with finite possibilities, such as the number of books borrowed from a library or the number of profile picture changes in a year. #### Characteristics of Discrete Probability Distributions: 1. **Countable Values**: Outcomes are recorded as whole numbers (e.g., 0, 1, 2, etc.). 2. **Randomness**: Outcomes depend on factors like personal preferences or external influences. ### Example Scenario: Profile Picture Changes In this study, we examine how often individuals change their profile pictures in a year. This is a discrete probability distribution because: - The number of changes is countable (e.g., 0, 1, 2). - Changes occur randomly, influenced by trends, preferences, or external factors. --- ## Objectives of the Study The study aims to: - Identify the number of times individuals change their profile pictures in a year. - Create a discrete probability distribution based on collected data. - Perform statistical analysis to interpret the results. --- ## Collected Data ### Summary of Findings The data was collected from **20 respondents**, and the frequency of profile picture changes was recorded. Below is the summary: | **Number of Changes** | **Frequency** | **Percentage** | |------------------------|---------------|----------------| | 0 | 9 | 45% | | 1 | 9 | 45% | | 2 | 2 | 10% | #### Key Observations: - **Most Common Frequency**: 45% of respondents (9 people) do not change their profile pictures at all. - **Least Common Frequency**: Only 10% of respondents (2 people) change their profile pictures twice a year. - **General Trend**: Most individuals rarely update their profile pictures. These findings suggest that profile picture changes are infrequent, with the majority of respondents either not changing them or doing so only once a year. --- ## Creating the Discrete Probability Distribution ### Step-by-Step Process 1. **Define the Random Variable**: Let $X$ represent the number of profile picture changes in a year. 2. **List Possible Outcomes**: The possible values for $X$ are: - $X = 0$: No changes - $X = 1$: One change - $X = 2$: Two changes 3. **Calculate Probabilities**: The probability of each outcome is calculated as: $$P(X = x) = \frac{\text{Frequency of } x}{\text{Total Respondents}}$$ Using the data: - $P(X = 0) = \frac{9}{20} = 0.45$ - $P(X = 1) = \frac{9}{20} = 0.45$ - $P(X = 2) = \frac{2}{20} = 0.10$ ### Probability Distribution Table | **Number of Changes ($X$)** | **Probability ($P(X)$)** | |-----------------------------|--------------------------| | 0 | 0.45 | | 1 | 0.45 | | 2 | 0.10 | Probabilities should always sum to 1. Verify this by adding $P(X = 0) + P(X = 1) + P(X = 2) = 0.45 + 0.45 + 0.10 = 1$. --- ## Statistical Analysis ### Mean (Expected Value) The **mean** of a discrete probability distribution is calculated using: $$E(X) = \sum [x \cdot P(X = x)]$$ Substitute the values: $$E(X) = (0 \cdot 0.45) + (1 \cdot 0.45) + (2 \cdot 0.10)$$ $$E(X) = 0 + 0.45 + 0.20 = 0.65$$ The expected number of profile picture changes per year is **0.65**, meaning most people change their profile pictures less than once annually. --- ### Variance and Standard Deviation The **variance** measures the spread of the data and is calculated as: $$\text{Var}(X) = \sum [(x - E(X))^2 \cdot P(X = x)]$$ Steps: 1. Compute $(x - E(X))^2$ for each $x$: - For $x = 0$: $(0 - 0.65)^2 = 0.4225$ - For $x = 1$: $(1 - 0.65)^2 = 0.1225$ - For $x = 2$: $(2 - 0.65)^2 = 1.8225$ 2. Multiply by $P(X = x)$: - For $x = 0$: $0.4225 \cdot 0.45 = 0.190125$ - For $x = 1$: $0.1225 \cdot 0.45 = 0.055125$ - For $x = 2$: $1.8225 \cdot 0.10 = 0.18225$ 3. Sum the results: $$\text{Var}(X) = 0.190125 + 0.055125 + 0.18225 = 0.4275$$ The **standard deviation** is the square root of the variance: $$\text{SD}(X) = \sqrt{\text{Var}(X)} = \sqrt{0.4275} \approx 0.653$$ Standard deviation helps us understand the variability in profile picture changes. A smaller SD indicates less variation among respondents. --- ## Key Takeaways - **Most Common Behavior**: People rarely change their profile pictures, with 90% of respondents changing them 0–1 times per year. - **Expected Value**: On average, individuals change their profile pictures **0.65 times per year**. - **Variability**: The standard deviation of **0.653** shows moderate consistency in behavior across respondents. --- ## Practice Problems 1. **Probability Calculation**: If 30 respondents were surveyed and 15 changed their profile pictures once, what is $P(X = 1)$? 2. **Expected Value**: Given the following probabilities, calculate $E(X)$: - $P(X = 0) = 0.3$ - $P(X = 1) = 0.5$ - $P(X = 2) = 0.2$ 3. **Variance**: Using the probabilities in Problem 2, calculate $\text{Var}(X)$. Many students forget to square the difference $(x - E(X))^2$ when calculating variance. Always double-check this step!