'X' is Gaussian random variable

Given a Gaussian random variable X with mean 0 and variance σ^2, we are asked to find the median of the variable Y, where Y is defined as the maximum of X and 0. The correct answer is 0.


To understand why the median of Y is 0, let's first define what a median is. In statistics, the median is the value that separates the higher half from the lower half of a dataset.


To find the median of Y, we need to understand its distribution. Y takes on the value of X when X is positive, and 0 when X is negative.

To visualize this, we can draw the probability density function (PDF) of Y. The PDF of Y will have a spike at 0, indicating that Y takes on the value of 0 with a certain probability, and it will have a distribution similar to X for positive values.


A Gaussian distribution, also known as a normal distribution, is symmetrical around its mean. This means that for every positive value of X, there is an equal probability of getting a negative value.

In our case, since X has a mean of 0, the probability of getting a positive value is the same as getting a negative value. Therefore, the area under the curve for positive and negative values of X is the same.


Since the area under the curve is symmetrical and the mean is 0, the median of X is also 0. This means that when X is positive, there is an equal probability of getting a value above or below 0.

Since Y takes on the value of 0 when X is negative, the median of Y is also 0. This is because the probability of getting a value above or below 0 is equal, and the median separates the higher and lower halves of the distribution.


In summary, the median of Y, which is defined as the maximum of X and 0, is 0. This is because the distribution of Y is symmetrical around 0, and the median separates the higher and lower halves of the distribution.