Probability for an exponential random distribution

[RET80]

Homework Statement

Find the exact value of P(|Y- μ| < 1.4σ) for an exponential random variable with parameter β.

Homework Equations

The only equation that I can think of is the exponential distribution equation:
∫(1/β)e^(-y/β)

The Attempt at a Solution

I have been unable to attempt it because I don't know what exactly this question is attempting to ask. It wants to solve for Parameter β, but it has a probability of P(|Y- μ| < 1.4σ)

Could I set my integral up from 1.4σ to infinity? and have my y variable equal to 'Y - μ' in my integral? I don't know what that would solve, if anything.

I'm not looking for an answer, just an idea of what to look for when attempting these equations, or better yet, what I'm missing or not understanding.

 

[RET80]

I have also noticed that it is in the structure of a Tchebysheff's inequality, which may or may not change something...

I attempted to do the integral of it. I took the integral from 0 to 1.4σ and β = Y - μ and ended up getting -1 + e ^(1.4σ / (Y - μ))

I have no idea if this is the correct way to get to this answer, but it is what I attempted to do.

 

[Stephen Tashi]

[itex] | Y - \mu | < 1.4 \sigma [/itex] implies
[itex] \mu-1.4\sigma < Y < \mu+1.4\sigma [/itex]
so that defines the interval for the integration.

Look up (or compute) the formula for [itex] \mu [/itex] and [itex] \sigma [/itex] in terms of the parameter [itex] \beta [/itex]. Substitute these expressions , either before or after you do the integral. The only "unknown" in the problem will then be [itex] \beta [/itex]. If this problem has a numerical answer then all the [itex]\beta[/itex]'s will cancel.

 

[RET80]

[itex] | Y - \mu | < 1.4 \sigma [/itex] implies
[itex] \mu-1.4\sigma < Y < \mu+1.4\sigma [/itex]
so that defines the interval for the integration.

Look up (or compute) the formula for [itex] \mu [/itex] and [itex] \sigma [/itex] in terms of the parameter [itex] \beta [/itex]. Substitute these expressions , either before or after you do the integral. The only "unknown" in the problem will then be [itex] \beta [/itex]. If this problem has a numerical answer then all the [itex]\beta[/itex]'s will cancel.

Well I did the integral from μ - 1.4σ to μ + 1.4σ using the above mentioned equation and I got:
[ -e^(-(μ + 1.4σ)/β) + e^ (-(μ - 1.4σ)/β)]

And it's looking for the exact answer of this.
Well the other equations I know are the equations for the mean and variance
given by:
Mean: μ = E(Y) = β
Variance: σ^2 = V(Y) = β^2

I don't know exactly what to do from here. I could do some substitution for the mean, but I don't know what that would do.

 

[Stephen Tashi]

I don't know exactly what to do from here. I could do some substitution for the mean, but I don't know what that would do.

You can substitute for [itex] \sigma [/itex] too.[itex] (\sigma^2 = \beta^2 [/itex] and because both are positive, we can conclude [itex] \sigma = \beta [/itex]. All the [itex] \beta's [/itex] will be gone when you reduce the fractions.

There is an important technicality about this problem. The formula for the exponential density only applies for those y that are non-negative. So when I said that the limit of integration was [itex] \mu - 1.46 \sigma [/itex] to [itex] [\mu + 1.46 \sigma [/itex], I was wrong. That lower limit would only be correct if

[itex] \mu - 1.46 \sigma \geq 0 [/itex].

i.e. [itex] \beta - 1.46 \beta \geq 0 [/itex] which is false.

The lower limit of integration should be 0 in this case.

If the problem asks about a smaller multiple of [itex] \sigma [/itex] such as "find the probability that [tex] | Y - \mu | < 0.5 \sigma [/tex] you could do it the way I originally suggested.