Expectation with log normal

In summary, the conversation discusses the expectation of log-normally distributed random variable X when x-a>0, and the speaker is looking for an analytical solution or good numerical approximation for E[log(x-a)]. The other speaker initially provides the definition of log-normal distribution and the expectation of log X, but then realizes they misread the question and apologizes for their previous response.
  • #1
econmath
2
0
What is the expectation, E(log(x-a)), when x is log normally distributed? Also x-a>0. I am looking for analytical solution or good numerical approximation.

Thanks
 
Mathematics news on Phys.org
  • #2
The random variable [itex]X[/itex] is said to be log-normally distributed if [itex]\log X[/itex] is normally distributed (I know, it's a weird naming convention). In other words, [itex]X= e^Z[/itex], where [itex]Z\sim \mathcal N(\mu_Z,\sigma_Z^2)[/itex], a normal random variable. So then [itex]\mathbb E [\log X]= E[Z] = \mu_Z[/itex].
 
  • #3
Yes of course, but I am looking for E[log (x-a)] not E[log(x)].

Thanks.
 
  • #4
Oh, yikes. I misread. Sorry for my useless answer.

I have no clue about your question.
 
  • #5
for your question. The expectation, E(log(x-a)), of a log normally distributed variable x with a shift parameter a and x-a>0 can be calculated analytically using the following formula:

E(log(x-a)) = ln(a) + (μ + σ^2/2)

Where μ and σ are the mean and standard deviation of the underlying normal distribution for x. This formula is derived from the properties of log normal distributions and can be found in many textbooks on probability and statistics.

Alternatively, a good numerical approximation for the expectation can be obtained using numerical integration techniques. This involves dividing the range of x-a into small intervals and calculating the integral of log(x-a) multiplied by the probability density function of the log normal distribution within each interval. The sum of these integrals will give a good approximation of the expectation.

I hope this helps. Please let me know if you have any further questions.
 

1. What is a log normal distribution?

A log normal distribution is a probability distribution of a random variable whose logarithm follows a normal distribution. It is often used to model data that is skewed to the right, meaning that there are a few large values and many small values.

2. How do you calculate the expectation of a log normal distribution?

The expectation of a log normal distribution can be calculated using the formula E[X] = e^(μ+σ^2/2), where μ is the mean and σ is the standard deviation of the corresponding normal distribution.

3. Can you have negative values in a log normal distribution?

No, a log normal distribution only includes positive values since the logarithm of a negative number is undefined.

4. What is the relationship between a log normal distribution and the central limit theorem?

The central limit theorem states that the sum of a large number of independent and identically distributed random variables will follow a normal distribution. Since the logarithm of a log normal distribution follows a normal distribution, this means that the log normal distribution is a result of the central limit theorem.

5. What are some real-world examples of data that can be modeled using a log normal distribution?

Some examples include income data, stock prices, population sizes, and the size of particles in a certain substance. Essentially, any data that is skewed to the right and has a few large values and many small values can be modeled using a log normal distribution.

Similar threads

Replies
1
Views
1K
Replies
24
Views
2K
  • General Math
2
Replies
44
Views
3K
  • General Math
Replies
3
Views
1K
Replies
10
Views
1K
  • General Math
Replies
2
Views
9K
  • General Math
Replies
6
Views
2K
Replies
4
Views
2K
  • General Math
Replies
3
Views
853
  • General Math
Replies
12
Views
921
Back
Top