If you roll a die, you get one of the following numbers: 1, 2, 3, 4, 5, 6. Each possibility occurs with equal probability of 1/6. The expected value of a dice roll is E(D) = 3.5 and the variance of a dice roll is Var(X) = 2.917. a) Suppose you roll a die and then add 1 to the roll to get a new random variable taking one of the following numbers: 2,3,4,5,6,7. What is the variance of this new random variable? b) Suppose you roll a die and then multiply the roll by 2 to get a new random variable taking one of the following numbers: 2, 4, 6, 8, 10, 12. What is the variance of this new random variable? c) Suppose you roll two dice and then take the sum of the two dice to get a number between 2 and 12. What is the variance of this sum random variable?
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!
If you roll a die, you get one of the following numbers: 1, 2, 3, 4, 5, 6. Each possibility occurs with equal probability of 1/6. The
a) Suppose you roll a die and then add 1 to the roll to get a new random variable taking one of the following numbers: 2,3,4,5,6,7. What is the variance of this new random variable?
b) Suppose you roll a die and then multiply the roll by 2 to get a new random variable taking one of the following numbers: 2, 4, 6, 8, 10, 12. What is the variance of this new random variable?
c) Suppose you roll two dice and then take the sum of the two dice to get a number between 2 and 12. What is the variance of this sum random variable?
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