What can we say about population mean from a sample size of 1? I am wondering what we can say, if anything, about the population mean, $\mu$ when all I have is one measurement, $y_1$ (sample size of 1).  Obviously, we'd love to have more measurements, but we can't get them.
It seems to me that since the sample mean, $\bar{y}$, is trivially equal to $y_1$, then $E[\bar{y}]=E[y_1]=\mu$.  However, with a sample size of 1, the sample variance is undefined, and thus our confidence in using $\bar{y}$ as an estimator of $\mu$ is also undefined, correct?  Would there be any way to constrain our estimate of $\mu$ at all? 
 A: Here is a brand-new article on this question for the Poisson case, taking a nice pedagogical approach:
Andersson. Per Gösta (2015). A Classroom Approach to the Construction of an Approximate Confidence Interval of a Poisson Mean Using One Observation. The American Statistician, 69(3), 160-164, DOI: 10.1080/00031305.2015.1056830.
A: If the population is known to be normal, a 95% confidence interval based on a single observation $x$ is given by $$x \pm 9.68 \left| x \right| $$
This is discussed in the article "An Effective Confidence Interval for the Mean With Samples of Size One and Two," by Wall, Boen, and Tweedie, The American Statistician, May 2001, Vol. 55, No.2. (pdf)
A: Sure there is. Use a Bayesian paradigm. Chances are you have at least some idea of what $\mu$ could possibly be - for instance, that it physically cannot be negative, or that it obviously cannot be larger than 100 (maybe you are measuring the height of your local high school football team members in feet). Put a prior on that, update it with your lone observation, and you have a wonderful posterior.
A: A small simulation exercise to illustrate whether the answer by @soakley works:
# Set the number of trials, M
M=10^6
# Set the true mean for each trial
mu=rep(0,M)
# Set the true standard deviation for each trial
sd=rep(1,M)
# Set counter to zero
count=0
for(i in 1:M){
 # Control the random number generation so that the experiment is replicable 
 set.seed(i)
 # Generate one draw of a normal random variable with a given mean and standard deviation
 x=rnorm(n=1,mean=mu[i],sd=sd[i])
 # Estimate the lower confidence bound for the population mean
 lower=x-9.68*abs(x)
 # Estimate the upper confidence bound for the population mean
 upper=x+9.68*abs(x)
 # If the true mean is within the confidence interval, count it in
 if( (lower<mu[i]) && (mu[i]<upper) ) count=count+1
}
# Obtain the percentage of cases when the true mean is within the confidence interval
count_pct=count/M
# Print the result
print(count_pct)
[1] 1

Out of one million random trials, the confidence interval includes the true mean one million times, that is, always. That should not happen in case the confidence interval was a 95% confidence interval. 
So the formula does not seem to work... Or have I made a coding mistake?
Edit: the same empirical result holds when using $(\mu, \sigma)=(1000,1)$;
however, it is $0.950097 \approx 0.95$ for $(\mu, \sigma)=(1000,1000)$ -- thus pretty close to the 95% confidence interval.
A: See Edelman, D (1990) 'A confidence interval for the center of an unknown unimodal distribution based on a sample size one' The American Statistician, Vol 44, no 4.
Article covers the Normal and Nonparametric cases.
