Integrate a Two-Sided Exponential Density

Question

The question asks the candidate to integrate a two-sided exponential density function, specifically to find the probability of a random variable defined by this density function within a specified range.

Codemia · Accepted Answer

The two-sided exponential distribution is defined by the probability density function (pdf) given by:

$$f(x) = 0.5 e^{-|x|}$$

for all real numbers x. This distribution is symmetric around zero and is often used in scenarios where events can occur in both directions from a central point.

To find the probability that a random variable X falls within the range of 1 to 3, we need to set up the integral of the pdf over this interval:

$$P(1 \leq X \leq 3) = \int_{1}^{3} f(x) \, dx = \int_{1}^{3} 0.5 e^{-x} \, dx$$

This integral will give us the area under the curve of the pdf from 1 to 3.

Now, we can compute the integral:

1. Calculate the antiderivative:
   $$\int 0.5 e^{-x} \, dx = -0.5 e^{-x} + C$$

2. Evaluate the definite integral:
   $$P(1 \leq X \leq 3) = [-0.5 e^{-x}]_{1}^{3} = -0.5 e^{-3} + 0.5 e^{-1}$$

3. Simplifying gives:
   $$P(1 \leq X \leq 3) = 0.5(e^{-1} - e^{-3})$$

The result of the integration provides the probability that the random variable X falls between 1 and 3. This probability can be interpreted in the context of the problem, indicating how likely it is for an event modeled by this distribution to occur within that range.

While performing this integration, we assume that the random variable X is continuous and that the two-sided exponential distribution accurately models the underlying process. If the distribution were not symmetric or had different parameters, the setup and calculations would need to be adjusted accordingly.

Integrate a Two-Sided Exponential Density

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