How to find pmf from cdf?
Understand the Problem
The question is asking how to derive the probability mass function (pmf) from the cumulative distribution function (cdf) for discrete random variables. This often involves taking the difference of the cdf values at successive points.
Answer
The formula for deriving PMF from CDF is $P(X = x) = F(x) - F(x - 1)$.
Answer for screen readers
The probability mass function (pmf) can be derived from the cumulative distribution function (cdf) using the formula:
$$ P(X = x) = F(x) - F(x - 1) $$
for discrete random variables.
Steps to Solve
- Identify the CDF Values
Start with the cumulative distribution function (CDF) of the discrete random variable, denoted as $F(x) = P(X \leq x)$.
- Calculate PMF from CDF
The probability mass function (PMF), denoted as $P(X = x)$, can be computed using the differences of the CDF values at successive points:
$$ P(X = x) = F(x) - F(x - 1) $$
This formula gives the probability that the discrete random variable (X) takes the value (x).
- Repeat for Each Discrete Point
If the CDF is defined for multiple discrete points, repeat the calculation for each point of interest to derive the corresponding PMF values.
The probability mass function (pmf) can be derived from the cumulative distribution function (cdf) using the formula:
$$ P(X = x) = F(x) - F(x - 1) $$
for discrete random variables.
More Information
Deriving the PMF from the CDF is fundamental in statistics, as it allows for a better understanding of the distribution of discrete random variables. The PMF represents the probabilities of specific outcomes, while the CDF provides the probabilities of outcomes less than or equal to a certain value.
Tips
- Confusing PMF with CDF: Ensure not to confuse the definitions. The PMF gives point probabilities, while the CDF gives cumulative probabilities.
- Using CDF values incorrectly: Remember to calculate the PMF using the right indices: $F(x)$ and $F(x - 1)$.
