Problem 4-D. Calculating Probability of an Event Defined by Two Continuous Random Variables An electronic gadget employs two integrated circuit chips: a signal processing chip and a power conditioning chip, which fail independently of each other. The gadget fails to operate only upon the failure of either one of the two IC chips (i.e., all other modalities of failure of the gadget can be ignored). The time to failure of a chip is defined as the time interval from its first use until its failure, and is random (i.e., varies from chip to chip). The time to failure for the signal processing chip, denoted by X, has an exponential distribution, having the probability density function fr(x) = a exp u(x) where u(x) is the unit step function, and a = 104 hr
Problem 4-D. Calculating Probability of an Event Defined by Two Continuous Random Variables An electronic gadget employs two integrated circuit chips: a signal processing chip and a power conditioning chip, which fail independently of each other. The gadget fails to operate only upon the failure of either one of the two IC chips (i.e., all other modalities of failure of the gadget can be ignored). The time to failure of a chip is defined as the time interval from its first use until its failure, and is random (i.e., varies from chip to chip). The time to failure for the signal processing chip, denoted by X, has an exponential distribution, having the probability density function fr(x) = a exp u(x) where u(x) is the unit step function, and a = 104 hr
Transcribed Image Text:Problem 4-D. Calculating Probability of an Event Defined by Two Continuous Random Variables
An electronic gadget employs two integrated circuit chips: a signal processing chip and a power
conditioning chip, which fail independently of each other. The gadget fails to operate only upon the failure
of either one of the two IC chips (i.e., all other modalities of failure of the gadget can be ignored). The
time to failure of a chip is defined as the time interval from its first use until its failure, and is random (i.e.,
varies from chip to chip). The time to failure for the signal processing chip, denoted by X, has an
exponential distribution, having the probability density function
where u(x) is the unit step function, and a = 10+ hr
The time to failure of the power conditioning chip, denoted by Y, is exponentially distributed, having the
fx(x) = a exp-* u(x)
probability density function
f; (y) = b exp- u(y)
-by
where b= 2× 10+ hr'
Find the probability that, when a given sample of gadget fails, the failure is due to the power conditioning
chip rather than the signal processing chip.
Ans. 2/3.
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