Distribution Function f(x)= .5e^|x|, find EX and Var(x)

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Discussion Overview

The discussion revolves around finding the expected value (EX) and variance (Var(X)) of a continuous random variable X with a specified density function. The focus is on the mathematical reasoning and calculations involved in determining these statistical measures.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • Post 1 presents the initial question regarding the expected value and variance of a random variable with the density function f(x) = 0.5e^|x|.
  • Post 2 challenges the validity of the density function, stating that it is not a proper density function as the integral over its range is not defined to equal 1, suggesting a correction to f(x) = 0.5e^{-|x|} instead.
  • Post 3 acknowledges the error and confirms the corrected density function f(x) = 0.5e^{-|x|}.
  • Post 4 provides an approach to calculating the expected value, suggesting that symmetry can be used to find E(X) without performing the integral, and outlines a method for calculating the variance using integration by parts.

Areas of Agreement / Disagreement

Participants initially disagree on the validity of the density function, but there is a subsequent agreement on the corrected function f(x) = 0.5e^{-|x|}. The discussion on how to compute EX and Var(X) remains open, with different methods proposed but no consensus reached on the calculations.

Contextual Notes

The discussion highlights the importance of correctly defining the density function, as the initial formulation was incorrect. There are also unresolved steps in the calculations for expected value and variance, particularly regarding the integration methods suggested.

Who May Find This Useful

This discussion may be useful for students or practitioners in statistics and probability theory, particularly those interested in properties of continuous random variables and their associated distributions.

badgerbadger
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Let X be a continuous random variable with density function

f(x)= .5e^|x|

for x range R. Find EX and Var(x)

help please!
 
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Check your problem again. That is NOT a density function. [itex]\int_{-\infty}^\infty f(x)dx[/itex] is not even defined, much less being 1. Did you mean [itex]f(x)= 0.5e^{-|x|}[/itex]?
 
haha yeah its f(x)= 0.5e^{-|x|}

sorry about that.
 
Then [itex]E(x)= \int_{-\infty}^{\infty}xe^{-|x|}dx[/itex] which you should be able to get by "symmetry" without needing to do the integral.

And then [itex]Var(X)= \int_{-\infty}^{\infty}x^2e^{-|x|}dx= 2\int_0^\infty x^2e^{x}dx[/itex] which you can do integrating by parts (twice).
 

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