Suppose a random variable X can take any value in the interval [−1,2] and a random variable Y can take any value in the interval [−2,3] .
a) The random variable X−Y can take any value in an interval [a,b] . Find the values of a and b :
a=
b=
b) Can the expected value of X+Y be equal to 6?
Yes or No
a).
a = -4
b = 4
b) No
a) The interval of X−Y can be found by subtracting the minimum value of Y from the maximum value of X and subtracting the minimum value of X from the maximum value of Y.
a = Max(X) - Min(Y) = 2 - (-2) = 4
b = Max(Y) - Min(X) = 3 - (-1) = 4
Therefore, a = 4 and b = 4.
b) Yes, the expected value of X+Y can be equal to 6. It is possible for the sum of the expected values of X and Y to equal 6.
a) To find the interval in which the random variable X - Y can take any value, we need to consider the maximum and minimum possible values of X - Y.
The maximum value of X - Y occurs when X is at its maximum value of 2 and Y is at its minimum value of -2. In this case, X - Y = 2 - (-2) = 4.
The minimum value of X - Y occurs when X is at its minimum value of -1 and Y is at its maximum value of 3. In this case, X - Y = -1 - 3 = -4.
Therefore, the random variable X - Y can take any value in the interval [-4, 4].
a = -4
b = 4
b) Yes, the expected value of X + Y can be equal to 6. The expected value of a sum of random variables is the sum of their individual expected values.
Let's denote the expected value of X as E(X) and the expected value of Y as E(Y).
Since X can take any value in the interval [-1, 2] and Y can take any value in the interval [-2, 3], the expected value of X is the average of the interval, which is (2 + (-1))/2 = 1/2, and the expected value of Y is the average of the interval, which is (3 + (-2))/2 = 1/2.
Therefore, the expected value of X + Y is E(X) + E(Y) = 1/2 + 1/2 = 1.
Since 1 is not equal to 6, the expected value of X + Y cannot be equal to 6.
No
a) To find the values of "a" and "b", we need to determine the range of possible values for the random variable X - Y.
To do this, we consider the extreme cases where X and Y take on their minimum and maximum values:
- When X is at its minimum (-1) and Y is at its maximum (3), X - Y = -1 - 3 = -4
- When X is at its maximum (2) and Y is at its minimum (-2), X - Y = 2 - (-2) = 4
So, the values of "a" and "b" such that X - Y can take any value in the interval [a, b] are:
a = -4
b = 4
b) To find if the expected value of X + Y can be equal to 6, we need to calculate the expected value and check if it is equal to the desired value.
The expected value of the sum of X and Y is given by E(X + Y) = E(X) + E(Y), where E(.) denotes the expected value.
Since X can take any value in the interval [-1, 2] and Y can take any value in the interval [-2, 3], the expected value of X is given by:
E(X) = (Minimum value of X + Maximum value of X) / 2 = (-1 + 2) / 2 = 1 / 2
Similarly, the expected value of Y is given by:
E(Y) = (Minimum value of Y + Maximum value of Y) / 2 = (-2 + 3) / 2 = 1 / 2
Therefore, the expected value of X + Y is:
E(X + Y) = E(X) + E(Y) = 1/2 + 1/2 = 1
Since the expected value of X + Y is equal to 1 and not 6, the answer is No.