Effortlessly determine the probability of a random variable falling within a specific range using our Cumulative Distribution Function on Calculator. Whether you are analyzing financial risk, scientific data, or academic problems, this tool provides instant, accurate results for Normal Distribution probabilities.
Cumulative Distribution Function on Calculator
Cumulative Distribution Function on Calculator Formula:
For a Normal Distribution, the CDF is defined as:
$$ \Phi(x) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{x – \mu}{\sigma \sqrt{2}} \right) \right] $$
Where $Z = \frac{x – \mu}{\sigma}$ is the standard score.
Formula Sources: Wikipedia, Wolfram MathWorldVariables:
- Mean (μ): The average value or the center of the distribution.
- Standard Deviation (σ): The measure of dispersion or spread of the data.
- Value (x): The threshold value for which you want to find the cumulative probability.
- Probability P(X ≤ x): The likelihood that a variable is less than or equal to $x$.
Related Calculators:
What is Cumulative Distribution Function on Calculator?
The Cumulative Distribution Function (CDF) describes the probability that a real-valued random variable $X$ with a given probability distribution will be found at a value less than or equal to $x$. In practical terms, it tells you the “area under the curve” from the far left up to your point $x$.
Using a Cumulative Distribution Function on Calculator is essential for statistical hypothesis testing, quality control in manufacturing, and predicting market fluctuations in finance. It simplifies complex calculus integrations into a few clicks, providing precise values for the Standard Normal Distribution.
How to Calculate Cumulative Distribution Function on Calculator (Example):
- Identify Parameters: Let Mean (μ) = 100 and Standard Deviation (σ) = 15.
- Select Target Value: We want to find the probability of a value being 115 or less ($x = 115$).
- Calculate Z-Score: $Z = (115 – 100) / 15 = 1$.
- Lookup/Calculate CDF: For $Z=1$, the area to the left is approximately 0.8413 or 84.13%.
Frequently Asked Questions (FAQ):
What is the difference between PDF and CDF? The Probability Density Function (PDF) shows the likelihood of a specific value, while the CDF shows the cumulative probability up to that value.
Can the CDF be greater than 1? No, the probability in a CDF always ranges from 0 to 1 (0% to 100%).
Is this calculator only for Normal Distribution? This specific module is optimized for the Normal Distribution, which is the most widely used distribution in statistics.
What is an inverse CDF? It is the value $x$ such that the cumulative probability is a specific number (e.g., finding the 95th percentile).