7.2.4. Properties of PDF

• Non-negativity: 𝑓(𝑥) ≥ 0

The property of non-negativity states that the Probability Distribution Function (PDF) must always produce values greater than or equal to zero. In simpler terms, when we talk about the likelihood of a random variable taking a certain value, we cannot have negative probabilities. This makes intuitive sense, as negative probability does not exist in real-life situations.

Imagine you're tossing a fair coin. The chance of getting heads is a probability of 0.5, and the chance of tails is also 0.5. If we were to obtain a negative probability (like -0.1), it would mean we're saying there's a 'chance' that you can't possibly get in real scenarios. Thus, probabilities being non-negative ensures that our model reflects a reality that is understandable and applicable.

• Normalization: ∫ 𝑓(𝑥)𝑑𝑥 = 1

Normalization refers to the requirement that the total probability represented by a PDF over all possible outcomes must equal one. This means that if we integrate the PDF from negative infinity to positive infinity, we should get a result of 1. This signifies that we are considering all possible outcomes of the random variable, which is essential for proper probability modeling.

Think about pouring a full bucket of water into different cups. The total amount of water (1 bucket = 100% probability) must be completely accounted for when dividing it among the cups (different potential outcomes). If we didn’t have normalization, it would be like claiming that the total amount of water exceeds or falls short of what exists, which wouldn’t make sense.

• Probability Calculation: 𝑃(𝑥 < 𝑋 < 𝑥) = ∫ 𝑓(𝑥)𝑑𝑥

This property describes how to calculate the probability that a random variable falls within a specific range, denoted by two values, x1 and x2. To find this probability, you integrate the PDF over the interval from x1 to x2. The area under the curve of the PDF between these two points represents the likelihood of the random variable taking on a value in that range.

Consider a scenario of measuring the heights of adult men. If you want to find the probability that a randomly selected man is between 5'9" and 6'1", you would look at the area under the height distribution curve between these two heights. Integrating gives you that area, which directly translates to the likelihood of your chosen height.

• Mean (Expected Value): 𝜇 = 𝐸[𝑋] = ∫ 𝑥𝑓(𝑥)𝑑𝑥

The mean or expected value of a random variable is a measure of the central tendency, representing what one can expect on average if an experiment is repeated many times. For a continuous random variable, the expected value is computed by integrating the product of the variable and its PDF. This gives a valuable reference point in the distribution.

If you think about your average scores in several tests, the expected value can be seen as the 'anchor point' where most of your scores center around. For example, if you were to assess your average score in mathematics over several semesters, gathering data from all your scores over those tests reveals insights about your general performance.

• Variance: 𝜎² = 𝐸[(𝑋−𝜇)²] = ∫ (𝑥 − 𝜇)²𝑓(𝑥)𝑑𝑥

Variance measures the spread or dispersion of the random variable's possible values relative to the mean. It is calculated by taking the expected value of the squared deviations from the mean. A high variance indicates that the values are spread out over a wide range, while a low variance indicates that they are closer to the mean.

Think of the heights of plants in a garden. If all plants are similar in height, the variance will be low, showing that the plants are close to a common height. However, if some plants are short and some are very tall, the variance will be high, indicating a wider range of sizes. This concept helps us understand how diverse we are among different measures.

--