# Distribution of a log-normal random variable with fixed dimension

Consider a random variable $$x_{ijk}$$, where $$ijk$$ is a subscript that indicates that the variable varies with 3 dimensions (e.g., firm, product, and country). $$x_{ijk}$$ is known to be i.i.d. log-normally distributed.

How to find the distribution of $$x_{ijk}$$ where $$k$$ is fixed? Is $$x_{ijk}$$ still an i.i.d. drawn sample and is $$x_{ijk}$$ still lognormally distributed?

• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Commented Mar 8, 2023 at 16:48

Due to the fact that your multivariate-normal distribution is independent (the covariance matrix is diagonal), it will be more intuitive to treat each dimension as being its 'own' random variable; i.e., let $$(I, J, K) \sim (\log(X), \log(Y), \log(Z))$$, where $$X, Y, Z$$ are all normal distributions with their own mean and variance parameters.
Now, due to the independence, the joint distribution is given by $$f(i, j, k) = f_I(i)f_J(j)f_K(k)$$, where $$f_A(a)$$ is the marginal density function of RV $$A$$. If you want to 'fix', aka condition on knowing $$k$$, then you want the distribution $$I, J|K$$. The density for this can be found with Bayes rule: $$f(i, j|k) = \frac{f(i, j, k)}{f_K(k)}\;;$$ but given we can factorise $$f(i, j, k)$$ using the above independence assumption we now have $$f(i, j|k) = \frac{f(i, j, k)}{f_K(k)} = \frac{f_I(i)f_J(j)f_K(k)}{f_K(k)} = f_I(i)f_J(j)\;.$$ So, you can see that the distribution of $$I, J|K$$ has the same distribution as the product of the two random variables $$I$$ and $$J$$. Putting this together, we can deduce that the distribution is still log-normally distributed (indeed, it is the product of two independent log-normal distributions) and they will still be independent.
• Maybe I misunderstood, but if your random variable $x$ depends on three things ($i,j,k$) then I took this to mean it is 3-dimensional? if it is a scalar, then you would have to be more clear on what it is you're doing, is it a scalar conditioned on these three values? Commented Mar 10, 2023 at 14:34