The manufacturer of NUKA soda has printed one of these four

The manufacturer of NUKA soda has printed one of these four letters, N,U,K,A in each bottle cap. The company offers a prize to anyone who collects all bottle caps that spell N-u-k-a. If all words are equally likely, And if yout get 1000 bottle caps. FInd the mean and variance of the distribution of the letters.

Solution

They are all equally likely....So, the prob. of occurence of each of them in a bottle = 1/4....

So, for a particular bottle...N,U,K,A can occur once with prob. 1/4...

Then,for 1000 samples...all can occur 250 times with a prob. 1/4.....

mean = 1000/4 =250...

variance = 4*[250^2 /4 ] - 250^2 = 0...