A cat's nightly catch of mice

Question

A cat catches a random number of mice each night, averaging $3$ but varying from night to night. Each mouse caught, independently of the others and of how many are caught, weighs an amount averaging $25$ grams.

What total catch weight should you expect on a given night?

Accepted Answer

Let N be the number of mice caught, and let the total weight be the sum of their weights. Both the count and the weights are random, so condition on N and use the tower rule: E[	ext{weight}] = E\big[\,E[	ext{weight} \mid N]\,\big]. Step 1, freeze the count. If N = n mice are caught, each averaging 25 grams, E[	ext{weight} \mid N = n] = 25\,n. Step 2, average over the count. Only E[N] = 3 is needed, since weights and count are independent: E[	ext{weight}] = E[25\,N] = 25\,E[N] = 25 	imes 3 = 75 	ext{ grams}. You expect 75 grams of mice: expected count (3) times average weight per mouse (…

A cat's nightly catch of mice

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