Let D1 be equal to the sum of the first three digits of your AUM student ID number, and let D2 be equal to the sum of the last three digits of your AUM student ID number and let D3 be equal to the sum of all the digits of your AUM student ID number
For example, if your student ID is 34251, then
D1 = 3+4+2= 7 and
D2 = 2+5+1 = 8
D3 = 3+4+2+5+1= 15
What are your D1, D2 and D3?
AUM ID=40862
D1=4+0+8=12 D2 =8+6+2=16 and D3 =4+0+8+6+2=20
Question 1 (30 points)
While walking from the car into your hotel, you dropped your engagement ring somewhere in the snow. The path is straight and D1 meters long. The density of its location seems to be constant along this route.
Delegate your assignment to our experts and they will do the rest.
What is the probability that the ring is within 3 meters away from your car? (6 points)
Working
Let X be random variable representing distance from case starting point;
Then X~ Uniform (0, D 1 )
Since D 1 =12, therefore, X~ Uniform (0, 12)
The pdf of x is give as:
P( =P( = P(
P(x) is evaluated as:
P( = dx
Evaluating the integral, we solve the value of 3 and 0 in the result
=
=0.25
What is the probability that the ring is more than 5 meters away from your car? (6 points)
Working
P( =P( = P( =
P(x) is evaluated as:
P( = dx
Evaluating the integral, we solve the value of 5 and 12 in the result
=
=
=0.5833
What is the probability that the ring is between 1.5 to 3.8 meters away from your car? (6 points)
Working
P( =P(
P( ) is evaluated as:
P( = dx
Evaluating the integral, we solve the value of 3.8 and 1.5 in the result
=
=0.3167-0.1250
=0.1917
What is the expected value of the distance from the car where the ring fell? (6 points)
Working
For a uniform distribution, the Expected value, E(x) is calculated as:
E(x)= where a=0 and b=12
=
=6 Meters
What is the standard deviation of the distance from the car where the ring fell? (6 points)
Working
Standard Deviation=
Standard Deviation= where a=0 and b=12
Standard Deviation=
=
=3.4641 Meters
Question 2 (35 points)
The lifetime of a server component purchased at Alibaba e-commerce is Exponentially distributed with a mean of D2 years.
What is the probability that the component fails in the first year? (7 points)
Working
D2=16
Since the distribution is exponential, the decay parameter k is calculated as; hence X~ Exp(k)
X~ Exp (0.0625)
The probability density function, pdf, is calculated as:
f(x)=
where, e=2.71828 and k=0.0625
=0.06059
=0
=0.06059-0
=0.0606
What is the probability that the component fails in between 6 and 15 years? (7 points)
Working
The probability density function, pdf, is calculated as:
f(x)=
where, e=2.71828 and k=0.0625
=0.6084
=0.3127
=0.0.6084-0.3127
=0.2957
What is the probability that the component fails in exactly 20 years? (7 points)
Working
f(x)=
P(x=20)=
=0.0625(0.286505)
=0.0179
What is the expected value of the lifetime of the server component? (7 points)
Working
Let X be a continuous random variable with an exponential density function with parameter m where m=16; the value of D2
By integrating using parts; u=mx and = so that and , we find that the expected value of X, E(X)=
=
=
Evaluate the limits in the integral:
=
E(X)=
What is the standard deviation of the lifetime of the server component? (7 points)
Working
By integrating by parts with and so that and , we have:
Replacing m=16, we have;
=
=
Variance(X)= -
= -
=
Standard deviation (X) =
=
Question 3 (35 points)
Your computer has D3 family photos. You decide to take 5 random photos (without replacement) from the collection (each is equally likely), to use as a slideshow on the TV when guests come over. Of your family photos, 3 are from your favorite vacation.
What is the probability that at least 2 photos in the slideshow are from your favorite vacation? (7 points)
Working
P(x)=
Where x= the number we are interested in coming from the group with A objects,
P(x) is the probability of x success, in n attempts in a population of N elements.
A=3, x=2, N=20 and n=5
P(
=1-0.8596
=0.1404
What is the probability that no photos in the slideshow are from your favorite vacation? (7 points)
Working
What is the probability that at most 3 photos in the slideshow are from your favorite vacation? (7points)
Working
=P(X=0) +P(X=1) + P(X=2) +(P(X=3)
What is the expected number of photos from your favorite vacation in the slideshow? (7 points)
Working
For a hypergeometric distribution;
Expected value, E(X)=n*p where p is the p proportion of photos obtained as and n is the sample size, n=5
E(X)=
E(X)=0.75
What is the standard deviation of the number of photos from your favorite vacation in the slideshow? (7 points)
Working
Since the distribution is a hypergeometric distribution with parameters;
N=20 as the population size,
N=5 as the sample size,
A=3 as the number of success in population
Standard deviation=
=
=
=0.7094
