Gaussian Random Variable
A Gaussian random variable, also known as a normal random variable, follows a normal distribution. It is characterized by its mean (μ) and variance (σ^2). In this case, the random variable X has a mean of 0 and a variance of 2.

Maximum Function
The maximum function, denoted as max(a, b), returns the larger value between a and b. In this case, the random variable Y is defined as the maximum of X and 0.

Probability Density Function (PDF)
The probability density function of a Gaussian random variable is given by:

f(x) = (1 / (σ * sqrt(2π))) * exp(-(x - μ)^2 / (2σ^2))

Where:
- σ is the standard deviation (sqrt(σ^2))
- π is the mathematical constant pi
- exp(z) represents the exponential function e^z

Determining the Median of Y
To determine the median of Y, we need to find the value y such that P(Y ≤ y) = 0.5. In other words, we are looking for the value y that divides the probability distribution of Y into two equal halves.

To find the probability distribution of Y, we consider two cases:
1. X ≥ 0: In this case, Y takes the value of X, so P(Y ≤ y) = P(X ≤ y). Since X follows a Gaussian distribution with mean 0 and variance 2, we can calculate this probability using the cumulative distribution function (CDF) of the Gaussian distribution.
2. X < 0:="" in="" this="" case,="" y="" takes="" the="" value="" of="" 0,="" so="" p(y="" ≤="" y)="P(0" ≤="" y)="1" if="" y="" ≥="" 0,="" and="" 0="" if="" y="" />< />

Calculating the Median
To calculate the median, we need to find the value y such that P(Y ≤ y) = 0.5. Let's consider the two cases separately:

1. X ≥ 0:
Using the CDF of the Gaussian distribution, we have:
P(X ≤ y) = Φ((y - μ) / σ)
Where Φ(z) is the CDF of the standard normal distribution. Since μ = 0 and σ = sqrt(2), the equation becomes:
P(X ≤ y) = Φ(y / sqrt(2))

2. X < />
P(Y ≤ y) = 0 if y < 0,="" and="" 1="" if="" y="" ≥="" />

To find the median, we need to solve the equation P(Y ≤ y) = 0.5. Since max(a, b) is a non-decreasing function, the median occurs when X = 0. Therefore, the median of Y is 0.

Answer: A) 0