You first roll a fair die once. Next you roll the die as many times as the outcome of this first roll. Let the random variable X be the total number of sixes in all the rolls of the die, including the first roll. Use conditional expectations to find the expected value of X. (express your answers up to 2 decimal places)
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
You first toss a fair coin five times. Next you toss the coin as many times as the number of heads showing up in these five tosses. Let the random variable X be the number of heads in all tosses of the coin, including the first five tosses. Use conditional expectations to find the expected value of X. (b) You first roll a fair die once. Next you roll the die as many times as the outcome of this first roll. Let the random variable X be the total number of sixes in all the rolls of the die, including the first roll. Use conditional expectations to find the expected value of X. (express your answers up to 2 decimal places)
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