How to find CDF from PMF?
Understand the Problem
The question is asking how to derive the cumulative distribution function (CDF) from the probability mass function (PMF). This involves summing the probabilities provided by the PMF for all outcomes up to a certain point to create the CDF.
Answer
To derive the CDF from the PMF, use $F(x) = \sum_{k \leq x} p(k)$.
Answer for screen readers
The cumulative distribution function (CDF) can be derived from the probability mass function (PMF) by summing the probabilities for all values up to a specified point $x$:
$$ F(x) = \sum_{k \leq x} p(k) $$
Steps to Solve
- Identify the Probability Mass Function (PMF)
Start with the given PMF, which provides the probability of each discrete outcome. For example, let $P(X = x) = p(x)$ represent the PMF.
- Define the Cumulative Distribution Function (CDF)
The CDF, denoted as $F(x)$, is defined as the sum of probabilities of the PMF for outcomes less than or equal to $x$.
- Sum the PMF probabilities
To find the CDF, sum the probabilities from the PMF for all outcomes less than or equal to $x$:
$$ F(x) = \sum_{k \leq x} P(X = k) = \sum_{k \leq x} p(k) $$
This means you will include all the probabilities from the PMF where $k$ is each value up to and including $x$.
- Example Calculation
If you have a PMF such as $P(X = 0) = 0.2$, $P(X = 1) = 0.5$, and $P(X = 2) = 0.3$, to find $F(1)$, you would calculate:
$$ F(1) = P(X = 0) + P(X = 1) = 0.2 + 0.5 = 0.7 $$
- Repeat for Other Values of x
Continue this process for other values of $x$ to get the complete CDF. For instance, for $F(2)$:
$$ F(2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.2 + 0.5 + 0.3 = 1.0 $$
The cumulative distribution function (CDF) can be derived from the probability mass function (PMF) by summing the probabilities for all values up to a specified point $x$:
$$ F(x) = \sum_{k \leq x} p(k) $$
More Information
The CDF is a crucial concept in probability and statistics, as it helps to understand the likelihood of a random variable being less than or equal to a certain value. Each point in the CDF is always non-decreasing and approaches 1 as $x$ approaches infinity.
Tips
- Forgetting to Use a Summation: One common mistake is to incorrectly compute the CDF by not summing the probabilities correctly or including only the last probability instead of all prior ones.
- Not Handling Discrete Outcomes Properly: Make sure to include all discrete outcomes for which the PMF is defined, especially when dealing with choices that aren't explicitly listed.
