what is sub gaussian random variable? what is its relationship with gaussian random variable ? also give se examples of sub gaussian random variable.
The Correct Answer and Explanation is :
Sub-Gaussian Random Variable: Definition and Relationship with Gaussian Random Variable
A sub-Gaussian random variable is a type of random variable whose tail probabilities decay at least as fast as those of a Gaussian (normal) distribution. Formally, a random variable ( X ) is called sub-Gaussian if there exists a positive constant ( C ) such that for all ( t \in \mathbb{R} ):
[
\mathbb{E}[e^{tX}] \leq e^{\frac{C^2 t^2}{2}}
]
for some constant ( C ), where ( \mathbb{E} ) represents the expectation operator.
This definition implies that sub-Gaussian random variables have light tails, meaning their probability of taking extreme values is exponentially small. They generalize Gaussian random variables in the sense that all Gaussian random variables are sub-Gaussian, but not all sub-Gaussian random variables are strictly Gaussian.
Relationship with Gaussian Random Variables
- Tail Bound: The probability of extreme deviations for a sub-Gaussian random variable is upper-bounded by that of a Gaussian distribution.
- Moment Conditions: If ( X ) is sub-Gaussian, then its moments (such as variance and higher moments) are controlled in a way similar to Gaussian variables.
- Hoeffding’s Lemma: Sub-Gaussian variables satisfy concentration inequalities similar to those for Gaussian variables, such as Hoeffding’s inequality.
- Stability Under Sums: Sums of independent sub-Gaussian random variables remain sub-Gaussian.
Examples of Sub-Gaussian Random Variables
- Gaussian Random Variable: A normal distribution ( X \sim N(\mu, \sigma^2) ) is inherently sub-Gaussian.
- Bounded Random Variables: Any random variable bounded in an interval (e.g., a uniform random variable on ([-1,1])) is sub-Gaussian.
- Rademacher Distribution: A symmetric Bernoulli random variable taking values ( \pm1 ) with equal probability is sub-Gaussian.
- Laplace Distribution: The Laplace distribution has exponential tails and is sub-Gaussian.
Now, generating an image to visually illustrate the concept of sub-Gaussian vs. Gaussian distributions.
Here is the generated image illustrating the comparison between Gaussian and sub-Gaussian distributions. Let me know if you need any modifications or further explanations!
