8.2. CDF for Discrete and Continuous Random Variables

If 𝑋 is a discrete random variable with probability mass function (PMF) 𝑝(𝑥), the CDF is:

𝐹(𝑥) = ∑𝑝(𝑡) 𝑡≤𝑥

Example: Let 𝑋 represent the result of rolling a fair six-sided die.

• 𝑝(𝑥) = $\frac{1}{6}$ for $x = 1,2,3,4,5,6$ • 𝐹(3) = 𝑃(𝑋≤ 3) = 𝑝(1)+ 𝑝(2)+ 𝑝(3) = $\frac{1}{2}$

In this chunk, we discuss discrete random variables and how to find their cumulative distribution function (CDF). A discrete random variable is one that can take on a finite number of values. The CDF for such variables is calculated using a probability mass function (PMF). Here's how it works:

1. The CDF, denoted as $F(x)$, gives the probability that the random variable $X$ is less than or equal to a particular value $x$.
2. To calculate $F(x)$ for a discrete variable, we sum up probabilities from the PMF for all values that are less than or equal to $x$. For example, when rolling a die, each side (1 through 6) has an equal probability of $\frac{1}{6}$.
3. Therefore, to find $F(3)$, we add the probabilities of rolling a 1, 2, or 3, resulting in $\frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$.

Imagine you are tracking how many friends come to your birthday party. You know that your friends either come or don't come, and you want to calculate the probability of a specific number coming. If you have 6 friends, you could find the probability of 0, 1, or 2 friends coming and then understand the cumulative probabilities as more friends decide to join. The idea is similar to rolling a die; you are summing the chances of rolling up to a specific number, like counting how many friends come to your party!

If 𝑋 is a continuous random variable with probability density function (PDF) 𝑓(𝑥), the CDF is:

𝐹(𝑥) = ∫ 𝑓(𝑡) 𝑑𝑡 −∞

Example: If 𝑓(𝑥)= 2𝑥 for 𝑥 ∈ [0,1], then:

𝐹(𝑥) = ∫ 2𝑡 𝑑𝑡 = [𝑡²]𝑥 = 𝑥² ₀ So, 𝐹(0.5) = (0.5)² = 0.25

This chunk explains how to find the cumulative distribution function (CDF) for continuous random variables. Continuous random variables can take any value within a given range. Here's the breakdown:

1. For continuous variables, the CDF is derived from the probability density function (PDF) by integrating the PDF over the desired range.
2. The formula $F(x) = \int_{-\infty}^{x} f(t) dt$ means that to find the cumulative probability up to $x$, we calculate the area under the PDF curve from negative infinity to $x$.
3. For instance, if the PDF is given by $f(x) = 2x$ for values between 0 and 1, we find $F(0.5)$ by integrating this function, resulting in a cumulative probability of 0.25.

Think of continuous random variables like measuring the amount of rain that falls in your town on any given day. If you want to know the likelihood that less than a certain amount, say 0.5 inches, falls in a day, you would consider all possible amounts of rain from 0 to 0.5 inches, adding up the probabilities across that range. This integration is like calculating the total area under a curve that represents rainfall probabilities.

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