Let X be the time in hours that the battery of a solar calculator functions properly between exposures to light sufficient to recharge it. Suppose the density of X is given by: f(x) = (50/6)x^-3 2 < x < 10. a) Verify that this is a valid continuous density. b) Calculate the expression for the cumulative distribution function of X and use it to calculate the probability that the charge of a randomly selected solar cell will last at most four hours before it needs to be used. at most four hours before it needs to be recharged. c) Calculate the average time that a battery charge lasts before it needs to be recharged. d) Calculate E[X2] and use its value to determine the variance of X. please only answer "d".
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!
Let X be the time in hours that the battery of a solar calculator
f(x) = (50/6)x^-3 2 < x < 10.
a) Verify that this is a valid continuous density.
b) Calculate the expression for the cumulative distribution function of X and use it to calculate the probability that the charge of a randomly selected solar cell will last at most four hours before it needs to be used.
at most four hours before it needs to be recharged.
c) Calculate the average time that a battery charge lasts before it needs to be recharged.
d) Calculate E[X2] and use its value to determine the variance of X.
please only answer "d".
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