The nth moment of a random variable X is defined to be the expectation E[Xn] of the nth power of X.
Recall that a Bernoulli random variable with parameter p is a random variable that takes the value 1 with probability p, and the value 0 with probability 1−p.
Let X be a Bernoulli random variable with parameter 0.7. Compute the expectation values of Xk, denoted by E[Xk], for the following three values of k: k=1,4, and 3203.
To compute the expectation values E[Xk] for the three values of k (1, 4, and 3203), we need to substitute the values of k into the formula for the expectation of a Bernoulli random variable.
Recall that for a Bernoulli random variable with parameter p, the expectation E[X] is given by:
E[X] = p
For k = 1:
E[X^1] = E[X] = p = 0.7
For k = 4:
E[X^4] = E[X^2 * X^2]
Since X^2 is also a Bernoulli random variable with parameter p, we can use the expectation formula to compute E[X^4].
E[X^4] = E[X^2] * E[X^2]
= (p^2) * (p^2)
= p^4
= (0.7)^4
= 0.2401
For k = 3203:
E[X^3203] = E[X^2 * X^3201]
Since X^2 is also a Bernoulli random variable with parameter p, we can use the expectation formula to compute E[X^3203].
E[X^3203] = E[X^2] * E[X^3201]
= (p^2) * E[X^3201]
To compute E[X^3201], we can break down the exponent as follows:
E[X^3201] = E[X^2 * X^3200]
= E[X^2] * E[X^3200]
= (p^2) * E[X^3200]
Continuing this process, we have:
E[X^3201] = E[X^2] * E[X^3200]
= (p^2) * E[X^3200]
= (p^2) * (p^3200)
= p^3202
= (0.7)^3202
Therefore, for k = 3203:
E[X^3203] = E[X^2] * E[X^3201]
= (p^2) * E[X^3201]
= (p^2) * (p^3202)
= p^3203
= (0.7)^3203
So, the expectation values E[X^k] for k = 1, 4, and 3203 are:
E[X^1] = 0.7
E[X^4] = 0.2401
E[X^3203] ≈ (0.7)^3203
To compute the expectation values E[Xk] for the given values of k, we need to use the formula for the expectation of a Bernoulli random variable.
The formula for E[Xk] of a Bernoulli random variable X with parameter p is:
E[Xk] = (0^k)(1-p) + (1^k)(p)
= (1-p) + p^k
Let's calculate the expectation values for the three values of k: k = 1, 4, and 3203.
For k = 1:
E[X1] = (1 - 0.7) + (0.7)^1 = 0.3 + 0.7 = 1
For k = 4:
E[X4] = (1 - 0.7) + (0.7)^4 ≈ 0.3 + 0.2401 ≈ 0.5401
For k = 3203:
E[X3203] = (1 - 0.7) + (0.7)^3203 ≈ 0.3 + 8.2857e-05 ≈ 0.300082857
Thus, the expectation values for Xk for the given values of k are:
E[X1] = 1
E[X4] ≈ 0.5401
E[X3203] ≈ 0.300082857