Question

13. A company produces light bulbs. Let X be a random variable that takes on the value 1 if a light bulb breaks within a year of being produced, and 0 otherwise. Let the probability that the light bulb breaks within a year be p. Let X1,...,Xn be i.i.d. draws from this distribution. (a) Derive the mean and variance of Xn := 1/n * sum_{i=1}^n Xi as a function of p and n alone. (b) What can be said about the distribution of [Xn - E(Xn)] / [var(Xn)]^1/2 when n is large? (c) Suppose you buy n = 100 light bulbs from this company. If p = 0.2, how likely is it that the proportion of lightbulbs that no longer work after one year is between 15 and 22? [Hint: you may use the result from part (b) to simplify your calculations.]

          13. A company produces light bulbs. Let X be a random variable that takes on the value 1 if
a light bulb breaks within a year of being produced, and 0 otherwise. Let the probability
that the light bulb breaks within a year be p. Let X1,...,Xn be i.i.d. draws from this
distribution.
(a) Derive the mean and variance of Xn := 1/n * sum_{i=1}^n Xi as a function of p and n alone.
(b) What can be said about the distribution of
[Xn - E(Xn)] / [var(Xn)]^1/2
when n is large?
(c) Suppose you buy n = 100 light bulbs from this company. If p = 0.2, how likely is
it that the proportion of lightbulbs that no longer work after one year is between 15
and 22? [Hint: you may use the result from part (b) to simplify your calculations.]

13. A company produces light bulbs. Let X be a random variable that takes on the value 1 if
a light bulb breaks within a year of being produced, and 0 otherwise. Let the probability
that the light bulb breaks within a year be p. Let X1,...,Xn be i.i.d. draws from this
distribution.
(a) Derive the mean and variance of Xn := 1/n * sumi=1^n Xi as a function of p and n alone.
(b) What can be said about the distribution of
[Xn - E(Xn)] / [var(Xn)]^1/2
when n is large?
(c) Suppose you buy n = 100 light bulbs from this company. If p = 0.2, how likely is
it that the proportion of lightbulbs that no longer work after one year is between 15
and 22? [Hint: you may use the result from part (b) to simplify your calculations.]

Added by Tyler K.

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