5. Suppose there are two manufacturers A and B of a smartphone battery. The factory A manufactures the fraction p E (0, 1) of all the batteries used in the smartphones while the factory B manufactures the rest. Assume that the lifetime of each battery is a continuous random variable T, parametrised by 1 > 0, with a density function: 0 0. What is the probability that your phone has a battery manufactured by A?

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5. Suppose there are two manufacturers A and B of a smartphone battery. The factory A
manufactures the fraction p e (0, 1) of all the batteries used in the smartphones while the factory
B manufactures the rest. Assume that the lifetime of each battery is a continuous random variable
Ta parametrised by 1 > 0, with a density function:
fa(t) = {7
0 <t < 2
0,
otherwise
The parameter a depends on the manufacturer of the battery. If the battery is manufactured by A
then 1 = 1 while if it is manufactured by B then 1 = 2.
Suppose you buy a new smartphone that is randomly picked among all the produced phones.
a. Compute the expectation and variance of the lifetime of the battery of your new phone?
b. Suppose the battery functions properly after time t> 0. What is the probability that your
phone has a battery manufactured by A?
Transcribed Image Text:5. Suppose there are two manufacturers A and B of a smartphone battery. The factory A manufactures the fraction p e (0, 1) of all the batteries used in the smartphones while the factory B manufactures the rest. Assume that the lifetime of each battery is a continuous random variable Ta parametrised by 1 > 0, with a density function: fa(t) = {7 0 <t < 2 0, otherwise The parameter a depends on the manufacturer of the battery. If the battery is manufactured by A then 1 = 1 while if it is manufactured by B then 1 = 2. Suppose you buy a new smartphone that is randomly picked among all the produced phones. a. Compute the expectation and variance of the lifetime of the battery of your new phone? b. Suppose the battery functions properly after time t> 0. What is the probability that your phone has a battery manufactured by A?
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