On any given day, mail gets delivered by either Alice or Bob. If Alice delivers it, which happens with probability 1/4, she does so at a time that is uniformly distributed between 9 and 11. If Bob delivers it, which happens with probability 3/4, he does so at a time that is uniformly distributed between 10 and 12. The PDF of the time X that mail gets delivered satisfies
a) fX(9.5)=
b) fX(10.5)=
9 10 11 `1
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There is an overlap between the two periods. Since each individual function is uniform, the resulting f(x) is a step function such that:
f(x): (1/4)/2 = 1/8 for x∈[9,10]
f(x): (1/4)/2+(3/4)/2 = 1/2 for x∈[10,11]
f(x): (3/4)/2 = 3/8 for x∈[11,12]
∫xf(x) = 1/8+1/2+3/8=1 checks.
a) 0.125
b) 0.5
a) fX(9.5) = "As a clown, I'm not very good at math, but let me clown around with this! Since Alice delivers with a probability of 1/4 and she does so between 9 and 11, we can say that the time she delivers follows a uniform distribution on the interval [9, 11]. For her to deliver at 9.5, the probability would be zero because it's not within her delivery window. So, fX(9.5) = 0. Don't worry, you won't have to wait around for a clown to deliver your mail."
b) fX(10.5) = "Let me clown around with this one too! Now, Bob delivers with a probability of 3/4, and he does so between 10 and 12. So, the time he delivers also follows a uniform distribution on the interval [10, 12]. For him to deliver at 10.5, the probability would be zero because it's not within his delivery window. So, fX(10.5) = 0. Looks like you won't be getting any mail at clown o'clock!"
To find the PDF (probability density function) of the time X that mail gets delivered, we need to consider the probabilities of Alice and Bob delivering the mail, as well as the distribution of the delivery times for each of them.
Let's start by calculating the probability that Alice delivers the mail, which is given as 1/4. This means that there is a 1/4 chance of Alice delivering the mail, and a 3/4 chance of Bob delivering it.
Next, we need to consider the distribution of the delivery times for Alice and Bob. Alice delivers the mail between 9 and 11, uniformly distributed. This means that the probability of Alice delivering the mail at any specific time between 9 and 11 is the same.
Since the delivery time is uniformly distributed, the probability density function for Alice's delivery time is simply a constant. Let's denote this constant as c1.
Similarly, Bob delivers the mail between 10 and 12, uniformly distributed. Again, the probability of Bob delivering the mail at any specific time between 10 and 12 is the same.
So, the probability density function for Bob's delivery time is also a constant. Let's denote this constant as c2.
To find the values of c1 and c2, we need to consider that the area under the probability density function should be equal to 1.
The total probability of Alice delivering the mail is 1/4, and the total probability of Bob delivering the mail is 3/4.
So, we have:
∫[9,11] c1dx = 1/4
∫[10,12] c2dx = 3/4
Integrating the above expressions, we get:
c1 * (11 - 9) = 1/4 => 2c1 = 1/4 => c1 = 1/8
c2 * (12 - 10) = 3/4 => 2c2 = 3/4 => c2 = 3/8
Now that we have the values of c1 and c2, we can determine the PDF of the time X that mail gets delivered.
a) fX(9.5)
To calculate the PDF at 9.5, we need to consider both Alice and Bob delivering the mail at this time.
The probability of Alice delivering the mail at exactly 9.5 is 0 since her delivery time is uniformly distributed.
The probability of Bob delivering the mail at exactly 9.5 is also 0 since his delivery time is uniformly distributed.
Therefore, the PDF at 9.5, fX(9.5), is 0.
b) fX(10.5)
To calculate the PDF at 10.5, we again need to consider both Alice and Bob delivering the mail at this time.
The probability of Alice delivering the mail at exactly 10.5 is 1/8, since her delivery time is uniformly distributed between 9 and 11.
The probability of Bob delivering the mail at exactly 10.5 is also 3/8, since his delivery time is uniformly distributed between 10 and 12.
Therefore, the PDF at 10.5, fX(10.5), is the sum of the probabilities of Alice and Bob delivering the mail at this time:
fX(10.5) = 1/8 + 3/8 = 1/2
So, the answers to the given questions are:
a) fX(9.5) = 0
b) fX(10.5) = 1/2