Find the mean, variance & a standard deviation of the binomial distribution with the given values of n and p.

n=126 p=0.27

A random variable following a binomial distribution over trials with success probability has PMF



Because it's a proper probability distribution, you know that the sum of all the probabilities over the distribution's support must be 1, i.e.



The mean is given by the expected value of the distribution,









The remaining sum has a summand which is the PMF of yet another binomial distribution with trials and the same success probability, so the sum is 1 and you're left with



You can similarly derive the variance by computing , but I'll leave that as an exercise for you. You would find that , so the variance here would be



The standard deviation is just the square root of the variance, which is