Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let be a $pdf$, $f(X)$, with exponential distribution and other $pdf$, $f(Y)$, with uniform distribution. I realized, 5000 times, for each one respectively that follow:

I used the R to generate a random sample of size $n = 5$, of the random variable $X$ (and $Y$) with parameter $\gamma = 2,0$ (for $Y$ I used min=0 e max = 4). I calculated the confidence interval 95% for $\mu$ using the data from this sample.

For $X$ variable I got a relative frequency closed to 80%, but for $Y$ variable I got a relative frequency closed to 95%. My question is Why with distribution uniform e small $n$ the confidence intervals have better behave?

share|cite|improve this question
Note that an exponential distribution is (a) right-skewed and (b) has a heavier tail than a normal distribution, with an excess kurtosis of 6. Both of these could affect the accuracy of a central limit theorem approximation for confidence intervals. – Henry Sep 3 '12 at 1:35
and uniform distribution? There is any formal proof for this? – juaninf Sep 3 '12 at 14:47
A uniform distribution is symmetric (implying $0$ skewness) and has an small negative excess kurtosis of $-\frac{6}{5}$ – Henry Sep 3 '12 at 19:20
There is any formal proof for this? – juaninf Sep 3 '12 at 19:35
You could just do the integrations, or look them up, e.g. at Wikipedia for an exponential distribution or a uniform distribution – Henry Sep 3 '12 at 20:16
up vote 2 down vote accepted

If you look at whether your confidence intervals are below $2$ or above $2$, of those from the exponential distribution about 11.3% are too low, 88.3% include the population mean of $2$, and 0.4% are too high, while with a normal distribution you would get about 2.5%, 95% and 2.5% respectively. You could use the following R code to simulate this.

The principal cause of this is that the exponential distribution is too skewed with a sample size of only $5$. The sample mean from an exponential distribution has a Gamma distribution which, (largely but not exclusively) because of its skewness, is poorly modelled by a Student's $t$ distribution.

cases    <- 5000
ss       <- 5
popmean  <- 2
#sampledata <- matrix(rnorm(cases*ss, mean=popmean, sd=1), ncol=ss)   #normal
#sampledata <- matrix(runif(cases*ss, min=0, max=2*popmean), ncol=ss) #uniform
 sampledata <- matrix(rexp(cases*ss, rate=1/popmean), ncol=ss)        #exp
samplemean  <- rowMeans(sampledata)
samplese    <- sqrt((rowSums(sampledata^2)/ss - samplemean^2)/(ss-1))
confintfactor <- qt(0.975, df=ss-1)               # about 2.776 for ss=5 
toolow  <- sum( samplemean + confintfactor*samplese  < popmean) / cases 
toohigh <- sum( samplemean - confintfactor*samplese  > popmean) / cases
c(toolow, 1-toolow-toohigh, toohigh)              # would like 0.025 0.95 0.025 

For the uniform distribution, about 3.3% of the confidence intervals are too low, 93.4% include the population mean of $2$, and 3.3% are too high. Here too the Student's $t$ distribution is not a perfect model, but it is better than before: in particular there is no skewness.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.