Example of distribution whose support is strictly positive I am looking for something similar to the normal distribution in which there are a mean and a standard deviation representing the amount of variation. However, the value must always be positive, i.e. $P(x \leq 0) = 0$.
 A: You may be interested in this gallery of distributions. In addition to


*

*the gamma distribution

*the lognormal distribution

*the $\chi^2$ distribution

*and the truncated normal distribution


that have already been brought up, you could check


*

*the F distribution

*the exponential distribution

*the Weibull distribution

*the power lognormal distribution


All of these have defined means and variances. Pick the one you like best.
A: What about the truncated normal distribution? Try for example
library(truncnorm)
x <- seq(0,10,by=.01)
plot(x,dtruncnorm(x, a=3, b=Inf, mean = 5, sd = 1),type="l")

This gives

By taking the mean of the underlying normal distribution $\mu$ (mean in the command) larger you can make it look "almost" normal, without a nonzero probability of nonpositive values.
A: There are infinitely many such distributions ... 
Consider the family of uniform distributions from $0$ to $N$ (non-inclusive of $0$), where $N$ is an arbitrary integer.
Now choose any one of these, say $X_1 \sim U(0,3)$. 
Then the sum of $X_1 + \dots + X_{10}$, where $X_1, \dots, X_{10} \overset{\text{iid}} \sim U(0,3)$ will be approximately normal in shape.
This uniform sum distribution is also known as the Irwin-Hall distribution.
A: Two suggestions for you:


*

*The Non-central $\chi^2_1$ distribution which is (can be) obtained by squaring the normal distribution $N(\mu,1)$? 




This easily satisfies your relation to normal as it always shares the parameters of some underlying normal. However, it always skews right, so while it may look sort of normal, it will never be.


*

*The distribution of number of heads from a series $n$ of coin flips (also known as the $Binomial(n,p)$ distribution) looks more and more like the normal distribution as $n$ increases.





The drawback here is that the support is always finite and includes 0; this latter can easily be remedied by just adding 1.
