Help with exponential distribution

In summary, the conversation discusses using R software to compute the mean of a sample from a population of 1000 independently and identically exponentially distributed variables with an expected value of 5. This is repeated for sample sizes of 20 and 100. The question is then raised about calculating the proportion of values of the mean that fall between 6.99 and 7.01. The response suggests considering what is known about the distribution of the sum of independent identically distributed variables.
  • #1
sneaky666
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X1,X2,...,XN are independently identically exponentially distributed with expected value of 5. How can I compute X[bar]n when n=20 and N=1000? Then compute the proportion of values of X[bar]n that lie between 6.99 and 7.01.

repeat the above question with n=100


My thoughts

so basically i am using code in R software to do this
and basically this question means that suppose there are 1000 iid exp. dist. with Expected value of each X is 5. Then get the mean of each X where the number of observations is 20 then 100.
i used
a=(1:1000) <--makes a vector [1,2,3,...,1000]
for(i in a){a=mean(rexp(20,5))} <---so 20 observations and expected value of 5, each slot in the vector gets replaced by a mean
plot(a)
and
a=(1:1000)
for(i in a){a=mean(rexp(100,5))}
plot(a)
i get the y-axis to be 0.1 to 0.35 for n=20 and 0.14 to 0.26 for n=100, what am I doing wrong? I don't get how to get a proportion of values of X[bar]n that lie between 6.99 and 7.01.
 
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  • #2
sneaky666 said:
and basically this question means that suppose there are 1000 iid exp. dist. with Expected value of each X is 5.

Correct, and before you start writing any code, think about what that statement implies.

For an exponential distribution, if you are told the mean you also know the variance.

What general results do you know about the distribution of the sum of independent identically distributed variables?
 

FAQ: Help with exponential distribution

1. What is an exponential distribution?

An exponential distribution is a probability distribution that describes the time between events occurring in a Poisson process. It is often used to model the time between occurrences of rare events.

2. How is an exponential distribution different from other distributions?

The exponential distribution is unique because it has a constant hazard rate, meaning the probability of an event occurring in a given time interval is always the same. This is in contrast to other distributions, such as the normal distribution, which have varying hazard rates.

3. What is the formula for calculating probabilities with the exponential distribution?

The formula for calculating probabilities with the exponential distribution is P(x < t) = 1 - e^(-λt), where λ is the rate parameter and t is the time interval.

4. How is the exponential distribution used in real life?

The exponential distribution is used in a variety of fields such as engineering, finance, and biology. It can be used to model the time between earthquakes, the lifespan of electronic components, and the time between customer arrivals in a queue, among other applications.

5. What are the limitations of the exponential distribution?

The exponential distribution assumes that events occur independently and at a constant rate, which may not always hold true in real-life situations. It also does not account for events that may occur simultaneously. Additionally, the exponential distribution is only suitable for modeling positive values and cannot be used for negative values.

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