. A man finds that two things cause him to wake up at night-noise or indigestion. If it is due to noise, the time to fall asleep again has exponential distribution with rate 5 per hour, and if it is due to indigestion the time to fall asleep again has exponential distribution with rate 2 per hour . When he is asleep, he stays asleep for an exponentially distributed time with mean 2 hours. Assuming the waking and sleeping perids are all independent of each other and that each time he wakes up it is due to noise with probability 2/3 and indigestion with probability 1/3, what proportion of time is he up because of indigestion and spend s ofn sleeping?
What on earth is sdu?
but it would take me a sleepless night to figure it out :(
Best answer I have seen in a week !!!
xyz
To find the proportion of time the man is up because of indigestion and the proportion of time he spends sleeping, we need to calculate the probabilities for the different scenarios.
Let's denote:
A: Waking up due to noise
B: Waking up due to indigestion
S: Sleeping
Given that each time he wakes up, it is due to noise with probability 2/3 and due to indigestion with probability 1/3, we can calculate the probabilities of waking up due to noise (P(A)) and waking up due to indigestion (P(B)):
P(A) = 2/3
P(B) = 1/3
We know that the time to fall asleep again has an exponential distribution with rate 5 per hour if it is due to noise and rate 2 per hour if it is due to indigestion.
Let:
X: Time to fall asleep again if waking up due to noise (exponential distribution with rate 5 per hour)
Y: Time to fall asleep again if waking up due to indigestion (exponential distribution with rate 2 per hour)
The average time spent sleeping is given as 2 hours.
If the man wakes up due to noise, the time to fall asleep again (X) has an exponential distribution with rate 5 per hour. Therefore, the expected time spent awake due to noise (E[X]) is 1/5 hours.
If the man wakes up due to indigestion, the time to fall asleep again (Y) has an exponential distribution with rate 2 per hour. Therefore, the expected time spent awake due to indigestion (E[Y]) is 1/2 hours.
The proportion of time the man spends sleeping is given by the average time spent sleeping (2 hours) divided by the sum of the expected times spent awake due to noise and indigestion:
Proportion of time spent sleeping = 2 / (E[X] * P(A) + E[Y] * P(B))
Now we can substitute the values and calculate the proportion of time spent sleeping:
Proportion of time spent sleeping = 2 / ((1/5 * 2/3) + (1/2 * 1/3))
Simplifying the expression:
Proportion of time spent sleeping = 2 / ((2/15) + (1/6))
Proportion of time spent sleeping = 2 / (4/30 + 5/30)
Proportion of time spent sleeping = 2 / (9/30)
Proportion of time spent sleeping = 2 * (30/9)
Proportion of time spent sleeping = 20/9
Therefore, the man spends 20/9 of his time sleeping.
To calculate the proportion of time he is up because of indigestion, we can subtract the proportion of time spent sleeping from 1:
Proportion of time up because of indigestion = 1 - Proportion of time spent sleeping
Proportion of time up because of indigestion = 1 - (20/9)
Proportion of time up because of indigestion = (9/9) - (20/9)
Proportion of time up because of indigestion = (9 - 20)/9
Proportion of time up because of indigestion = -11/9
However, a negative proportion does not make sense in this context, so the answer is not meaningful. It is important to note that there might be an error in the calculations or assumptions made.