Expectation of a function of a continuous random variable

In summary, the expected value of a function of a continuous random variable X, g(X), is equal to the integral of g(x) multiplied by the probability density function of X, fX(x), over the entire domain of X. This still holds even if W, the resulting variable, is discrete. In this case, the integral can be replaced by a sum over the values of W. Whether W itself is discrete or continuous does not affect this formula.
  • #1
kingwinner
1,270
0
If W=g(X) is a function of continuous random variable X, then E(W)=E[g(X)]=

∫g(x) [fX(x)] dx
-∞
============================
Even though X is continuous, g(X) might not be continuous.

If W happens to be a discrete random variable, does the above still hold? Do we still integrate ∫ (instead of sum ∑)?

Does it matter whether W itself is discrete or continuous?

Thanks!
 
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  • #2
If the variable is discrete you can still express in terms of integrals using dirac delta functions though it is simpler to replace the integral with a sum over values.
 
  • #3
kingwinner said:
If W=g(X) is a function of continuous random variable X, then E(W)=E[g(X)]=

∫g(x) [fX(x)] dx
-∞
============================
Even though X is continuous, g(X) might not be continuous.

If W happens to be a discrete random variable, does the above still hold?
Yes. Consider the simplest case, where g(x) = 1 if x is in [a,b] and 0 otherwise. Then

[tex]\begin{align*}
E[W] &= 1\cdot P(W = 1) + 0\cdot P(W = 0)\\
&= P(g(X) = 1)\\
&= P(X\in [a,b])\\
&= \int_a^b f_X(x)\,dx.
\end{align*}[/tex]

On the other hand, the formula also gives

[tex]\int_{-\infty}^\infty g(x)f_X(x)\,dx = \int_a^b f_X(x)\,dx.[/tex]
 
  • #4
Go for summations...that will solve ur problem and rest all may complicate
 

What is the expectation of a function of a continuous random variable?

The expectation of a function of a continuous random variable is a measure of the central tendency or average value of the function over all possible values of the continuous random variable. It is calculated by taking the integral of the function multiplied by the probability density function of the continuous random variable.

How is the expectation of a function of a continuous random variable calculated?

The expectation of a function of a continuous random variable is calculated by taking the integral of the function multiplied by the probability density function of the continuous random variable. This can be represented by the formula E[g(X)] = ∫g(x)f(x)dx, where g(x) is the function and f(x) is the probability density function.

What is the relationship between the expectation of a function of a continuous random variable and the expected value of the continuous random variable itself?

The expectation of a function of a continuous random variable is related to the expected value of the continuous random variable itself by the formula E[g(X)] = g(E[X]), where E[X] is the expected value of the continuous random variable X. In other words, the expectation of a function can be thought of as the function of the expected value.

Why is the expectation of a function of a continuous random variable important in statistics?

The expectation of a function of a continuous random variable is important in statistics because it allows us to calculate the average value of a function over all possible values of the continuous random variable. This can be used to make predictions, estimate parameters, and evaluate the performance of statistical models.

How is the expectation of a function of a continuous random variable used in real-world applications?

The expectation of a function of a continuous random variable is used in a variety of real-world applications, such as finance, economics, and engineering. It is particularly useful in decision making, risk assessment, and optimization problems. For example, in finance, the expectation of a function can be used to determine the expected return on an investment or to price financial derivatives.

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