Final Answer:
1. The probability density of x₁ + x₂ follows a normal distribution with a mean of 2μ and a variance of 2.
2. The maximum likelihood estimate (MLE) of μ is the average of the sample mean, which is (x₁ + x₂)/2.
Step-by-step explanation:
The sum of two independent normal random variables is also a normal distribution. If x₁ and x₂ are both sampled from a normal distribution with mean μ and variance σ²=1, then x₁ + x₂ follows a normal distribution with a mean equal to the sum of their individual means (μ + μ = 2μ) and a variance equal to the sum of their individual variances (1 + 1 = 2). Therefore, the probability density of x₁ + x₂ is a normal distribution with a mean of 2μ and a variance of 2.
Regarding the maximum likelihood estimate (MLE) of μ, for a normal distribution, the MLE is simply the sample mean. In this case, the sample mean is (x₁ + x₂)/2, as you're averaging the values from the random sample. This estimator is unbiased and efficient for the population mean μ. Thus, the MLE for μ is (x₁ + x₂)/2.
To summarize, the sum x₁ + x₂ follows a normal distribution with a mean of 2μ and a variance of 2. The maximum likelihood estimate for μ, based on the sample mean, is (x₁ + x₂)/2.