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Explore the probability of a continuous random variable using a free probability density function calculator. Let’s dive deeper to understand the concept and see how this tool can save time and provide accurate results.
In statistics:
“A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a specific range of values.”
It is also known as a probability distribution function.
Note: The integral of a PDF over its entire range is always equal to 1.
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This curve shows how probabilities are distributed. A free PDF calculator allows you to compute values quickly and visualize the curve easily.
The beta distribution is defined by two shape parameters:
$$ B(x, y) = \int t^{x-1} (1-t)^{y-1} dt $$
$$ f(x; a, b) = \frac{1}{B(a, b)} x^{a-1} (1-x)^{b-1} $$
a, b: shape parameters
B: beta function
Used to test relationships between categorical variables:
$$ f(x; k) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2} $$
k: degrees of freedom
Γ: gamma function
Used to compare variances of two datasets:
$$ f(x; d_1, d_2) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x B\left(\frac{d_1}{2}, \frac{d_2}{2}\right)} $$
d1, d2: degrees of freedom
B: beta function
Represents equal probability across a range:
$$ f(x) = \begin{cases} \frac{1}{b-a}, & a \le x \le b \\ 0, & \text{otherwise} \end{cases} $$
a: lower bound
b: upper bound
x: value to evaluate
Used when the population variance is unknown:
$$ f(t) = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\sqrt{v \pi} \, \Gamma\left(\frac{v}{2}\right)} \left(1 + \frac{t^2}{v}\right)^{-\frac{v+1}{2}} $$
v: degrees of freedom
Γ: gamma function
Special normal distribution with mean 0 and standard deviation 1:
$$ \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2} $$
For any mean and standard deviation:
$$ f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$
μ: mean
σ: standard deviation
Find the PDF of a normal distribution where:
Solution:
$$ f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$
Substitute values:
$$ f(24) = \frac{1}{\sqrt{2*3.14*2^2}} e^{-\frac{(24-3.3)^2}{2*2^2}} $$ $$ f(24) \approx 1.093 \times 10^{-24} $$
| Cumulative Distribution Function (CDF) | Probability Density Function (PDF) |
|---|---|
| Probability that a variable ≤ x | Likelihood of a variable being exactly x |
In probability, k represents the number of successes in n trials (binomial coefficient). Learn more using our binomial distribution calculator.
Probability is the area under the PDF curve. Probability density is the likelihood at a specific point. Use our probability calculator for computations.
NORMDIST calculates normal distribution probabilities given mean and standard deviation.
It represents the 97.5th percentile of a standard normal distribution. Check percentiles using our percentile calculator.
Probability distributions describe the likelihood of outcomes in a dataset. Using a PDF calculator helps quickly analyze distributions, variances, and probabilities for continuous data.
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