The life length of a light bulb is a random variable X (measured in months) that

Question

The life length of a light bulb is a random variable X (measured in months) that is normally distributed with mean 2 and variance 1. The life lengths of light bulbs are independent and identically distributed (IID) random variables. You rent a small studio for a five month period. The studio has a socket for only one light bulb. Suppose that you want to buy all your light bulbs at the beginning of the lease. You start with no bulb in the studio and replace a light bulb with a new one as soon as a bulb burns out. (a) You buy n light bulbs and denote their individual life lengths by the IID random variables X_1, X_2, ..., X_n. Compute the mean and the variance of the total life length of the n bulbs. (b) You buy 2 bulbs. What is the probability that they will last for the entire 5-month period?

Explanation / Answer

a) mean of n number of light bulbs =2*n=2n

and variance =1*n=n

b) here mean of 2 bulbs =2*2=4

and std deviation =(2)1/2 =1.414

henc eP(X>5) =1-P(X<5)=1-P(Z<(5-4)/1.414)=1-P(Z<0.707)=1-0.7602=0.2398

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