Diagnostic Test In an angiogram test, the doctor injects a...

Diagnostic Test In an angiogram test, the doctor injects a dye into your artery and then performs an $X$ -ray to view blockages. Assuming that the dye is excreted after a time $T$ (in hours) which is a random variable that is exponentially distributed with $a=1 / 3$ (a) Find the mean time for the dye to be excreted. (b) Find the probability that the time for the dye to be excreted is less than 90 minutes.

Diagnostic Test In an angiogram test, the doctor injects a dye into your artery and then performs an $X$ -ray to view blockages. Assuming that the dye is excreted after a time $T$ (in hours) which is
a random variable that is exponentially distributed with $a=1 / 3$
(a) Find the mean time for the dye to be excreted.
(b) Find the probability that the time for the dye to be excreted
is less than 90 minutes.

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Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between occurrences of events, particularly when the events occur continuously and independently at a constant average rate. Its key feature is the memoryless property, meaning that the probability of an event occurring in the future does not depend on how much time has already elapsed.

Mean of Exponential Distribution

For an exponential distribution, the mean (or expected value) is determined by the reciprocal of the rate parameter (lambda). This value represents the average time expected until the occurrence of the event in question.

Unit Conversion

Unit conversion is an essential concept in probability and statistics, especially when the given parameters and required outputs are expressed in different units. Ensuring consistency in units, such as converting minutes to hours, is critical for accurate computation of probabilities and expected values.

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