# Probability- Exponential Random Variables

• Roni1985
In summary, the conversation discusses the use of large deviation methodology to find a lower bound for the rate function R(a) when X1, X2... are independent and identically distributed mean 1 exponential random variables. The equation R(a) \leq \frac{-logP[Sn >n*a]}{n} is mentioned, and the attempt at a solution involves finding the probability using the pdf of a Gamma distribution, but the integral is difficult to solve. Help is requested for proceeding with finding the limit as n tends to infinity for R(a).
Roni1985

## Homework Statement

Suppose X1,X2... are iid mean 1 exponential random variables. Use large deviation methodology to give a lower bound for the rate function R(a) for a>1

## Homework Equations

R(a) $$\leq$$ $$\frac{-logP[Sn >n*a]}{n}$$

## The Attempt at a Solution

I know that a sum of exponential random variables is Gamma (n, 1).

I'm having a problem with finding the probability.

the pdf of a gamma dist random variable is:$$\frac{x^(^n^-^1^)*e^(^-^x^)}{(n-1)!}$$
I think after transforming it, this is the pdf.

But I'm getting a very hard integral and even my TI-89 can't solve it.
I think I'm doing something wrong.

Last edited:
I know that the rate function is the limit as n tends to infinity of R(a) so I don't know how to proceed.Any help would be greatly appreciated!

## 1) What is an exponential random variable?

An exponential random variable is a type of continuous random variable that represents the time between events in a Poisson process. It is often used to model the time between occurrences of rare events.

## 2) How is an exponential random variable different from other types of random variables?

Unlike other types of random variables, an exponential random variable has a constant rate of change over time. This means that the probability of an event occurring at any given time is independent of all previous events.

## 3) What is the probability density function for an exponential random variable?

The probability density function (PDF) for an exponential random variable is given by f(x) = λe^(-λx), where λ is the rate parameter and x is the time. This function describes the probability of an event occurring at a specific time.

## 4) How is the mean of an exponential random variable calculated?

The mean of an exponential random variable is equal to 1/λ, where λ is the rate parameter. This means that as the rate parameter increases, the mean decreases, and vice versa.

## 5) Can an exponential random variable have a negative value?

No, an exponential random variable cannot have a negative value. This is because it represents the time between events, and time cannot be negative. The range of an exponential random variable is from 0 to infinity.

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