The amount of time, T, that the effect of caffeine is felt on your body after drinking a cup of coffee is given by a continuous random variable with an exponential distribution. On average, caffeine is felt for about 5 hours after drinking the cup. (Answer part a-c) a. What is the probability that the effect lasts at least 4 hours? b.  What is the probability that the effect lasts fewer than 6 hours? c.  You usually go to sleep at 11:30 pm, and drink exactly one cup of coffee a day. Suppose you wish to guarantee at most a 5% chance of feeling the effects of caffeine when you go to sleep. What's the latest that you should drink your only cup of coffee?

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The amount of time, T, that the effect of caffeine is felt on your body after drinking a cup of coffee is given by a continuous random variable with an exponential distribution. On average, caffeine is felt for about 5 hours after drinking the cup. (Answer part a-c)

a. What is the probability that the effect lasts at least 4 hours?

b.  What is the probability that the effect lasts fewer than 6 hours?

c.  You usually go to sleep at 11:30 pm, and drink exactly one cup of coffee a day. Suppose you wish to guarantee at most a 5% chance of feeling the effects of caffeine when you go to sleep. What's the latest that you should drink your only cup of coffee?

Expert Solution
Step 1

Introduction:

For an exponentially distributed random variable X, with mean λ, the cumulative distribution function (cdf) at the point X = x is:

Statistics homework question answer, step 1, image 1

Here,

Statistics homework question answer, step 1, image 2

Thus, the cdf is:

Statistics homework question answer, step 1, image 3

Step 2

(a)

The probability that the effect lasts at least 4 hours is 0.4493, as calculated below:

Statistics homework question answer, step 2, image 1

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