How to calculate conditional expectation E[g(x) | x>= Q] for x ~ exp(1)

In summary, the expectation of $g(x)$ for $x\geq Q$ can be found by taking the integral of $g(x)$ times the probability density function $f(x)$ over the interval $[Q, \infty)$, which is exactly what you have done.
  • #1
user_01
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Given that $X$ is exponentially distributed continuous random variable $X\sim \exp(1)$ and $g(x)$ is as below. How can I find the Expectectaion of $g(x)$ for the condition that $x\geq Q$, i.e. $\mathbb{E}[g(x)\ | \ x\geq Q]$.

$$g(x) = \frac{A}{\exp(-bQ+c)}\Big(\frac{1 + \exp(-bQ+c)}{1 + \exp(-dx+c)}-1\Big)$$

I suppose I should start by considering an event $\Phi$ such that $\Phi = \mathbb{P}[x \geq Q]$. However, I don't know how to go around this condition.

All constants are positive real values.
 
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  • #2
Is the following solution correct for the above question? If it is OK, then I have found the solution. But I will really appreciate if someone can let me know if the following method is correct.

$$\mathbb{E}[g[x]] = \int_Q^\infty g(x). f(x) dx$$

$$\mathbb{E}[g[x]]= \int_Q^\infty\frac{A}{\exp(-bQ+c)}\Big(\frac{1 + \exp(-bQ+c)}{1 + \exp(-dx+c)}-1\Big) e^{-x} dx$$
 
  • #3
Hi user_01,

Yes, your proposed solution is correct. The relevant theorem here is known as the Law of the Unconscious Statistician (LOTUS). See here.
 

FAQ: How to calculate conditional expectation E[g(x) | x>= Q] for x ~ exp(1)

1. What is conditional expectation?

Conditional expectation is a statistical concept that measures the expected value of a random variable given certain conditions or information. It is denoted as E[X | Y], where X is the random variable and Y is the condition or information.

2. How do you calculate conditional expectation?

To calculate conditional expectation, you need to multiply the probability of each possible outcome of the random variable with its corresponding value and then sum them up. This can be written as E[X | Y] = ∑ x P(X = x | Y) * x, where x represents each possible outcome of the random variable.

3. What is the formula for calculating conditional expectation for a continuous random variable?

The formula for calculating conditional expectation for a continuous random variable is E[X | Y] = ∫ x f(x | Y) dx, where f(x | Y) is the conditional probability density function of the random variable X given the condition Y.

4. How do you calculate conditional expectation for a specific condition, such as x >= Q?

To calculate conditional expectation for a specific condition, such as x >= Q, you need to first find the conditional probability density function f(x | x >= Q) and then use the formula E[X | x >= Q] = ∫ x f(x | x >= Q) dx. In the case of a continuous random variable, you can also use the formula E[X | x >= Q] = ∫ Q ∞ f(x) dx.

5. Can you provide an example of calculating conditional expectation for a continuous random variable with a condition?

Yes, for example, if we have a continuous random variable X that follows an exponential distribution with a rate parameter λ = 1, and we want to calculate E[X | X >= 2], we first need to find the conditional probability density function f(x | X >= 2). This can be written as f(x | X >= 2) = f(x) / P(X >= 2), where f(x) is the probability density function of X and P(X >= 2) is the probability of X being greater than or equal to 2. Then, we can use the formula E[X | X >= 2] = ∫ 2 ∞ f(x | X >= 2) dx = ∫ 2 ∞ f(x) / P(X >= 2) dx. By substituting the values, we get E[X | X >= 2] = ∫ 2 ∞ (1/2) * e^(-x) dx = 1/2.

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