Expected Value of X^3 – Definition | Calculation | Examples

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In this article, we will explain how to calculate the expected value of X^3 where X can be any continuous or discrete random variable. For a discrete random variable, the expected value is calculated by multiplying the cube of the random variable by the pdf and summing up over all possible values of x. For a continuous random variable, it is found by integrating the x^3f(x) over the domain of f(x).

Expected Value of X^3 for a Discrete Random Variable:

Let X be a discrete random variable with probability mass function p(x). Then the required expected value can be calculated using the formula, E(X^3) = \sum_{x}x^3p(x)

Example:

Consider a discrete random variable X whose probability mass function is given as follows,

In order to calculate the required value, we first create columns to calculate the values of x^3 and x^3p(x).

We now sum up the values of the last column as required by the formula, \begin{align*}E(X^3) &= \sum_{x}x^3p(x) \\ &= (-0.8) + (-0.2) +0 + 0.3 + 0.8 \\ &= 0.1\end{align*}

Expected Value of X^3 for a Continuous Random Variable:

Let X be a continuous random variable with probability distribution function f(x). Then the required expected value can be calculated using the formula, E(X^3) = \int_{x}x^3f(x)dx

Example:

Let X be a continuous random variable with pdf of X is given as follows, f(x) = 2x \text{ where } 0 < x < 1

We now calculate the expected value of x^3 as follows, \begin{align*}E(X^3) &= \int_{x}x^3f(x) dx \\ &= \int_{0}^{1}2x^5 dx &= [2x^6/5]_0^1 = 1/3 \end{align*}.

Thus the required expected value is 1/3.

Does the expected value of x^3 equal mu_3?

The expected value of X^3 refers to the third raw moment of the random variable whereas the symbol \mu_3 (mu_3) refers to the third central moment of the random variable. Thus we conclude that these two quantities are not equal to each other.

The third raw moment is denoted by the symbol \mu_3^{'} (mu_3 prime). Thus we conclude that the expected value of X^3 is equal to \mu_3^{'}.

Ways to get the Expected Value of X^3:

1. One way to get the expected value is to use the formula above and obtain the expected value by integrating the value of x^3f(x) over the possible values of X.
2. Another way to calculate the required expected value is by using the moment-generating function of the random variable. If we differentiate the moment generating function three times and then put t=0 we obtain the required expected value. For example, suppose X follows an exponential distribution with a mean equal to 1, that is,  x~exp(1). The moment generating function of the exponential distribution is given as, M_X(t) = \frac {1}{1 - \lambda t} The value of the parameter is equal to 1. Differentiating thrice we get, \frac{d^3}{dt^3}[M_X(t)] = \frac {24}{(1 - t)^4} Putting t=0 we get that the required answer is 24.

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