Random variables X and Y are both normally distributed with mean 100 and standard
deviation 4. It is known that random variable X+Y is also a normal distribution.
a. What is the mean of X+Y?
b. What is the standard deviation of X+Y?
I see that the mean is 200 but I don't know how it arrived to.
To find the mean of the sum of two random variables X and Y, we can simply add the means of X and Y. In this case, both X and Y have a mean of 100.
a. The mean of X+Y = mean of X + mean of Y = 100 + 100 = 200.
Therefore, the mean of X+Y is 200.
b. To find the standard deviation of the sum of two random variables X and Y, we can use the formula:
Standard deviation of X+Y = √(standard deviation of X^2 + standard deviation of Y^2).
Given that X and Y both have a standard deviation of 4, we can plug in the values into the formula:
Standard deviation of X+Y = √(4^2 + 4^2) = √(16 + 16) = √32 ≈ 5.657.
Therefore, the standard deviation of X+Y is approximately 5.657.
To find the mean of the sum of two random variables, we can simply add their means together. Since both random variables X and Y have a mean of 100, the mean of the sum X+Y is:
Mean(X+Y) = Mean(X) + Mean(Y) = 100 + 100 = 200
Therefore, the mean of X+Y is 200.