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Is the random variable Y a lognormal variable?
The random variable Y is lognormally distributed with parameters μ and σ. This is the standard definition, but notice that the parameters are specified as the mean and standard deviation of X = log (Y). Recently, a SAS customer asked me an interesting question. What if you know the mean and variance of Y, rather than log (Y)?
What is the sample variance of one observation?
The sample variance is s2 = 0, meaning there is no variation. If we increase the last value to 10, the sample variance is s2 = .36. Increasing the last observation to 12, s2 = 1.45, and increasing it to 14, s2 = 3.3.
How to calculate the sample mean and variance?
In words, the sample variance is computed by subtracting the sample mean from each observation and squaring. Then you add the results and divide by n − 1, the number of observations minus 1. For the data at hand, the calculations can be summarized as follows:
Is there a way to simulate the lognormal distribution?
The answer is yes. In terms of μ and σ, the mean of Y is m = exp (μ + σ 2 /2) and the variance is v = (exp (σ 2) -1) exp (2μ + σ 2 ). You can invert these formulas to get μ and σ as functions of m and v. Wikipedia includes these formulas in its article on the lognormal distribution, as follows: Let’s rewrite the expression inside the logarithm.
Can you compute the usual parameters for log ( y )?
Mathematically, the situation is that if m and v are the mean and variance, respectively, of a lognormally distributed variable Y, can you compute the usual parameters for log (Y)? The answer is yes. In terms of μ and σ, the mean of Y is m = exp (μ + σ 2 /2) and the variance is v = (exp (σ 2) -1) exp (2μ + σ 2 ).
Which is the sampling distribution of a normal variable?
Sampling Distribution of a Normal Variable . Given a random variable . Suppose that the X population distribution of is known to be normal, with mean X µ and variance σ 2, that is, X ~ N (µ, σ). Then, for any sample size n, it follows that the sampling distribution of X is normal, with mean µ and variance σ 2 n, that is, X ~ N µ, σ n .