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Question Number 8232 by FilupSmith last updated on 03/Oct/16

Every day, for n days, you put either  $1, $2, or $3 into a saving account.  It is random as to how much you save  each day. What is the average amount  you will have saved in n days?

$$\mathrm{Every}\:\mathrm{day},\:\mathrm{for}\:{n}\:\mathrm{days},\:\mathrm{you}\:\mathrm{put}\:\mathrm{either} \\ $$$$\$\mathrm{1},\:\$\mathrm{2},\:\mathrm{or}\:\$\mathrm{3}\:\mathrm{into}\:\mathrm{a}\:\mathrm{saving}\:\mathrm{account}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{random}\:\mathrm{as}\:\mathrm{to}\:\mathrm{how}\:\mathrm{much}\:\mathrm{you}\:\mathrm{save} \\ $$$$\mathrm{each}\:\mathrm{day}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:{average}\:\mathrm{amount} \\ $$$$\mathrm{you}\:\mathrm{will}\:\mathrm{have}\:\mathrm{saved}\:\mathrm{in}\:{n}\:\mathrm{days}? \\ $$

Answered by prakash jain last updated on 03/Oct/16

Assuming uniform distribution.  On an average you will save=((1+2+3)/3)=$2  every day.  So you will save 2n dollars after n days.

$$\mathrm{Assuming}\:\mathrm{uniform}\:\mathrm{distribution}. \\ $$$$\mathrm{On}\:\mathrm{an}\:\mathrm{average}\:\mathrm{you}\:\mathrm{will}\:\mathrm{save}=\frac{\mathrm{1}+\mathrm{2}+\mathrm{3}}{\mathrm{3}}=\$\mathrm{2} \\ $$$$\mathrm{every}\:\mathrm{day}. \\ $$$$\mathrm{So}\:\mathrm{you}\:\mathrm{will}\:\mathrm{save}\:\mathrm{2}{n}\:\mathrm{dollars}\:\mathrm{after}\:\mathrm{n}\:\mathrm{days}. \\ $$

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