Final answer:
The mean and variance of Y when Y = 4X + 3 are 43 and 96, respectively. For Y = -2X - 3, the mean and variance are -23 and 24, respectively. These transformations demonstrate how linear operations on a random variable will affect its distribution's mean and variance.
Step-by-step explanation:
To calculate the mean and variance of the new random variable Y, we utilize the properties of linear transformations on random variables.
For Y = 4X + 3, the mean of Y, E(Y), can be found by multiplying the mean of X by 4 and then adding 3, so E(Y) = 4E(X) + 3. Given that E(X) = 10, E(Y) = 4(10) + 3 = 40 + 3 = 43.
The variance of Y, Var(Y), is found by squaring the coefficient of X (which is 4 in this case) and then multiplying by the variance of X, so Var(Y) = 4²Var(X). Since Var(X) = 6, Var(Y) = 4²(6) = 16(6) = 96.
Now, for Y = -2X - 3, the mean of Y is E(Y) = -2E(X) - 3, which gives E(Y) = -2(10) - 3 = -20 - 3 = -23. The variance does not change with addition or subtraction, and therefore Var(Y) is simply Var(Y) = (-2)²Var(X) = 4(6) = 24.
The mean and variance of Y are crucial for understanding its distribution. They define its center and the degree to which the values of Y are spread out around the mean.