Let X be physical distance, measured in inches, from its anchoring point where a bicycle spoke will snap. Then X has the following probability density function: f(x)= (6/169)x(x-x/13) for 0 ≤ x ≤ 13 and 0 otherwise. a)What is the probability that a bicycle spoke snaps less than 3 inches from its anchoring point? b) What is the probability that X >6? c)What is the probability that 2< X < 8? d)What is the expected value of X (E(X))? e)What is the expected value of x2 f) What is the variance of X?
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 physical distance, measured in inches, from its anchoring point where a bicycle spoke will snap. Then X has the following probability density
a)What is the probability that a bicycle spoke snaps less than 3 inches from its anchoring point?
b) What is the probability that X >6?
c)What is the probability that 2< X < 8?
d)What is the
e)What is the expected value of x2
f) What is the variance of X?
g) What is the standard deviation of X?
h)What is the probability that X is less than its expected value?
i)What is the expected value of X^2
j)What is the 60th percentile of X?
k) What is the probability that X is within 2 standard deviations of its expected value?
l) What is the probability that X = 8?
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