Right-skewed distribution with mean equals to mode? Is it possible to have a right-skewed distribution with mean equal to mode? If so, could you give me some example? 
 A: If the distribution is discrete, sure. It's easy. For example, a distribution with probability mass function


*

*$P(X=0) = 0.36$ 

*$P(X=1) = 0.40$ 

*$P(X=2) = 0.13$ 

*$P(X=3) = 0.10$ 

*$P(X=4) = 0.01$
is right (i.e. positively) skewed and has both a mean and a mode of 1. 
A: Easy examples come from binomial distributions -- which can hardly be dismissed as pathological or as bizarre counter-examples constructed ad hoc. Here is one for 10 trials and probability of success 0.1. Then the mean is 10 $\times$ 0.1 = 1, and 1 also is the mode (and for a bonus the median too), but the distribution is manifestly right skewed. 
The code giving the number of successes 0 to 10 and their probabilities 0.348678... and so forth is Mata code from Stata, but your favourite statistical platform should be able to do it. (If not, you need a new favourite.) 
: (0::10), binomialp(10, (0::10), 0.1)
                  1             2
     +-----------------------------+
   1 |            0   .3486784401  |
   2 |            1    .387420489  |
   3 |            2   .1937102445  |
   4 |            3    .057395628  |
   5 |            4    .011160261  |
   6 |            5   .0014880348  |
   7 |            6    .000137781  |
   8 |            7   8.74800e-06  |
   9 |            8   3.64500e-07  |
  10 |            9   9.00000e-09  |
  11 |           10   1.00000e-10  |
     +-----------------------------+

Among continuous distributions, the Weibull distribution can show equal mean and mode yet be right-skewed.  
