If the second and third moment about the origin are 8 and 18 and the third moment about mean is -14, then the first moment about the origin is:

The correct answer is 2.

Key Points

Here, it is already given that the second and third moments about the origin are 8 and 18 respectively.

The nth moment about the origin, E(X^n), is defined as the expectation of the nth power of the random variable X. The nth centered moment (or the nth moment about the mean), E((X - E[X])^n), is the expectation of the nth power of the difference of the random variable X and its expectation E[X].

The nth moment about the mean can also be expressed in terms of moments about the origin.

Based on our problem statement, we have:

• E(X^2) = 2nd moment about the origin = 8
• E(X^3) = 3rd moment about the origin = 18
• E((X - E[X])^3) = 3rd moment about the mean = -14

We can utilize the following relationship between the moments about the origin and the moments about the mean:

E((X - E[X])^n) = E[X^n] - n * E[X] * E[X^(n-1)] + n(n-1)/2 * E[X^(n-2)] * (E[X])^2 - ...

For n = 3, the above equation simplifies to:

E((X - E[X])^3) = E[X^3] - 3*E[X]E[X^2] + 2(E[X])^3.

Substitute the given values into the above equation:

-14 = 18 - 3* E[X]8 + 2(E[X])^3.

Solving this linear equation for E[X] (the first moment about the origin, also known as the expectation or mean of X), we find that E[X] = 2.

So the first moment about the origin would be 2, hence the option 4) 2 is the correct answer.