Function of gaussian random variable

In summary, a gaussian random variable is a continuous random variable that follows a normal distribution and is described by its mean and standard deviation. Its function is to describe the probability distribution of a continuous random variable and it is unique in its symmetry and bell-shaped curve. Some practical applications of a gaussian random variable include modeling natural phenomena, financial systems, and making predictions in statistical analysis and machine learning. The probability of a specific value for a gaussian random variable can be calculated by finding the area under the curve of the normal distribution at that point.
  • #1
Jply
3
0
I'm having trouble showing the following relation:

E(exp(z)) = exp(E(z^2)/2)

where z is a zero-mean gaussian variable and E() is the avg

anyone can help?
 
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  • #2
Expectation. Look up the definition of the expectation of f(z), where z is some random variable, and do the integral it gives you. Although you don't have to actually do the integration as such, just show the two integrals are equal.
 
  • #3
Thanks a lot
 

1. What is a gaussian random variable?

A gaussian random variable is a continuous random variable that follows a normal distribution. It is described by its mean and standard deviation, and is often used to model real-world phenomena in statistics and probability.

2. What is the function of a gaussian random variable?

The function of a gaussian random variable is to describe the probability distribution of a continuous random variable. This allows us to make predictions and draw conclusions about the behavior of a system or phenomenon.

3. How is a gaussian random variable different from other distributions?

A gaussian random variable is unique in that it is symmetric and bell-shaped, with the majority of values falling within one standard deviation of the mean. This makes it a versatile tool for modeling a wide range of natural phenomena.

4. What are some practical applications of a gaussian random variable?

Gaussian random variables are commonly used in fields such as finance, engineering, and physics to model complex systems. They are also frequently used in statistical analysis and machine learning algorithms to make predictions based on data.

5. How can we calculate the probability of a specific value for a gaussian random variable?

The probability of a specific value for a gaussian random variable can be calculated by finding the area under the curve of the normal distribution at that point. This can be done using mathematical formulas or statistical software.

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