List of formulas related to continuous random variable and its functions:
- Continuous random variable ( X ):
- This is a quantity that can take any value from a certain interval.
- Distribution Function (CDF) ( F(x) ):
- Defined as:
[
F(x) = P(X \leq x)
] - This is the probability that a random variable ( X ) will take on a value less than or equal to ( x ).
- Probability Density (PDF) ( f(x) ):
- It is related to the distribution function as follows:
[
f(x) = \frac{d}{dx} F(x)
] - This is the derivative of the distribution function, which shows what is the probability that a random variable will take a value in the neighborhood of ( x ).
- Properties of the distribution function:
- ( F(x) ) is always in the range from 0 to 1:
[
0 \leq F(x) \leq 1
] - ( \lim_{x \to -\infty} F(x) = 0 )
- ( \lim_{x \to +\infty} F(x) = 1 )
- Properties of probability density:
- The probability density is always non-negative:
[
f(x) \geq 0
] - The integral of the probability density over the entire space is 1:
[
\int_{-\infty}^{+\infty} f(x) \, dx = 1
]