Let X be a uniform random variable on the range {−1,0,1,2}. Let Y=X^4. Use the expected value rule to calculate E[Y].
E[Y]=
Answer:
E[Y] = 4.5
Well, isn't this a random party! Let's calculate the expected value of Y using the expected value rule. Since X is a uniform random variable and can take on the values -1, 0, 1, or 2, we need to calculate the expected value of Y for each of these values and then take their average.
For X = -1, we have Y = (-1)^4 = 1.
For X = 0, we have Y = 0^4 = 0.
For X = 1, we have Y = 1^4 = 1.
For X = 2, we have Y = 2^4 = 16.
Now, let's calculate the weighted average by multiplying each value of Y by its respective probability and adding them up:
E[Y] = (-1) * P(X = -1) + 0 * P(X = 0) + 1 * P(X = 1) + 16 * P(X = 2).
Since X is a uniform random variable, each value has an equal probability of occurring, which is 1/4:
E[Y] = (-1) * (1/4) + 0 * (1/4) + 1 * (1/4) + 16 * (1/4) = (-1/4) + 0 + 1/4 + 16/4 = 16/4 = 4.
So, the expected value of Y is 4. Isn't that a square surprise?
To calculate E[Y], we need to use the expected value rule. The expected value of a function of a random variable can be found by taking the sum of the function values multiplied by their respective probabilities.
In this case, we have the random variable X which is uniformly distributed on the set {−1,0,1,2} and Y=X^4.
To find E[Y], we first need to determine the probability distribution of X. Since X is a uniform random variable, all values in its range have equal probabilities. In this case, each value in the range {−1,0,1,2} has a probability of 1/4.
Now, let's calculate E[Y] using the expected value rule:
E[Y] = Σ (y * P(Y=y)), where the summation is taken over all possible values of Y.
Let's calculate the value of Y for each possible value of X and multiply it by the probability of that value occurring:
For X = -1, Y = (-1)^4 = 1
P(X=-1) = 1/4
Contribution to E[Y]: 1 * (1/4) = 1/4
For X = 0, Y = 0^4 = 0
P(X=0) = 1/4
Contribution to E[Y]: 0 * (1/4) = 0
For X = 1, Y = (1)^4 = 1
P(X=1) = 1/4
Contribution to E[Y]: 1 * (1/4) = 1/4
For X = 2, Y = (2)^4 = 16
P(X=2) = 1/4
Contribution to E[Y]: 16 * (1/4) = 4
Summing up the contributions to E[Y]:
E[Y] = (1/4) + 0 + (1/4) + 4 = 4.5
Therefore, E[Y] = 4.5.