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Integrate a Two-Sided Exponential Density

Goldman SachsData Scientist · Online Assessment
EasyData ScientistOnline AssessmentStatistics & Math

Last updated

This question tests the ability to integrate probability density functions (PDFs) over specified intervals, specifically handling the absolute value function in symmetric distributions.

Problem Statement

Consider a continuous random variable X with a probability density function defined as:

f(x) = 0.5 * exp(-|x|)

where |x| is the absolute value of x, and the domain is all real numbers (-infinity < x < infinity).

Your task is to calculate the probability P(1 < X < 3). Provide your answer as a mathematical expression and, if possible, a decimal approximation.

Example

  • To find P(-1 < X < 1), you would compute the integral of f(x) from -1 to 1.
  • Due to symmetry, this is equal to 2 * integral of (0.5 * exp(-x)) dx from 0 to 1.
  • 2 * [0.5 * (-exp(-x))] from 0 to 1 = 1 - exp(-1) approx 0.632.

Constraints

  • You must demonstrate the step-by-step integration process.
  • Pay close attention to how the absolute value affects the sign of the exponent for positive and negative values of x.

Hints

Hint 1

Recall that the absolute value |x| is equal to x when x > 0 and -x when x < 0.

Hint 2

Since the interval [1, 3] is entirely in the positive domain, the expression simplifies to 0.5 * exp(-x).

Hint 3

The integral of 0.5 * exp(-x) is -0.5 * exp(-x); apply the Fundamental Theorem of Calculus over the bounds 1 to 3.

Follow-up questions

  • What is the cumulative distribution function (CDF) for this random variable for x > 0?
  • How would the integration change if the interval included zero, such as P(-2 < X < 2)?
  • Is this distribution leptokurtic or platykurtic compared to a standard normal distribution?