a. The 68% of the widget weights lie between 33 ounces and 55 ounces.
b. The percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c. The percentage of widget weights lying above 22 ounces is approximately = 5%.
Based on the given data points: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, the five-number summary can be identified as follows:
Minimum: 5
First Quartile (Q1): 7
Median (Q2): 11
Third Quartile (Q3): 15
Maximum: 18
Now, let's answer the questions related to the distribution of widget weights using the Empirical Rule:
a) The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data lies within one standard deviation of the mean. Since the mean is 44 ounces and the standard deviation is 11 ounces, we can say that approximately 68% of the widget weights lie between 44 - 11 = 33 ounces and 44 + 11 = 55 ounces.
b) To determine the percentage of widget weights lying between 11 and 55 ounces, we need to calculate the z-scores for these values. The z-score is calculated using the formula: z = (x - mean) / standard deviation.
For 11 ounces: z1 = (11 - 44) / 11 = -33/11 = -3
For 55 ounces: z2 = (55 - 44) / 11 = 11/11 = 1
Using the Empirical Rule, we know that approximately 68% of the data lies within one standard deviation of the mean. Therefore, the percentage of widget weights lying between 11 and 55 ounces is approximately 68%.
c) To determine the percentage of widget weights lying above 22 ounces, we need to calculate the z-score for 22 ounces: z = (22 - 44) / 11 = -22/11 = -2.
Using the Empirical Rule, we know that approximately 95% of the data lies within two standard deviations of the mean. Therefore, the percentage of widget weights lying above 22 ounces is approximately 100% - 95% = 5%.
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Jay's hair grows about 8 inches each year. Write a function that describes the length l in inches that Jay's hair will grow for each year k. Which kind of model best describes the function?
Answer:
l=8k
Step-by-step explanation:
Answer:
y=8k
Step-by-step explanation:
PLEASE HELP!!! I’ll mark brainliest
Answer: I think it’s the first choice. Foreign Language, Math/CS, and social studies
Step-by-step explanation:
9. A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is 4 minutes. Interarrival times and service times are exponential, and the system may be modeled as an M/M/s/GD/[infinity]/[infinity] queuing system. It costs $20 per hour to operate a cash register, and a cost of 25¢ is assessed for each minute the customer spends in the cash register area. How many registers should the store open? [1 point]
10. On a sunny Spring day MiniGolf has a revenue of $2,000. If the day is cloudy, the revenue drops by 20%. A rainy day reduces the revenue by 80%. If today’s weather is sunny, there is an 80% chance that tomorrow’s weather will be sunny with no chance of rain. If it is cloudy, there is 20% chance that tomorrow will be rainy and 30% chance that it will be sunny. Rain will continue through the next day with a probability of 0.80, but there is 10% chance that it might be sunny. Determine (a) the expected daily revenue for the MiniGolf. (b) the average number of days the weather will not be sunny. [1 point]
The expected number of days with non-sunny weather is given as:
E(x) = P(1) + 2P(2)
= 0.26 + 2 × 0.204
= 0.668.
Hence, this is the required answer.
1 Register should be kept open.
(10)a. The expected daily revenue for the MiniGolf $1,920.
(10)b. The average number of days the weather will not be sunny is 0.668.
9. Calculation of the number of registers:
Given that
λ = 18 customers/hour
µ = 1 / 4 min
= 15 customers/hour
So,
λ / µ = 18 / 15
= 1.2
It is given that s is the number of servers we are looking for.
Let's start solving the queuing system:
For calculating the optimal number of cash registers we have the formula;
rho = λ / (s × µ)
Where,ρ = Traffic Intensity
λ = Arrival Rate
µ = Service Rate
The number of Servers (s) is calculated as:
s = ρλ / µ(If s is fractional, round it up to the nearest integer)
Calculation of ρ;
ρ = λ / (s × µ)
= 18 / (s × 15)
For the above equation let's put some values of s to find the value of ρ;s ρ=λ/(s×µ)
18/(15s)=1.2/1.8
= 0.67, rounded up to 1.1.
The register should be kept open.
10. Calculation of expected daily revenue:
(a) Given that revenue on a sunny day is $2,000.
If it's cloudy, the revenue decreases by 20%, and if it's rainy, the revenue decreases by 80%.
Therefore, Revenue on a cloudy day
= $2,000 - 20% of $2,000
= $2,000 - $400
= $1,600
Revenue on a rainy day = $2,000 - 80% of $2,000= $2,000 - $1,600 = $400
Now, We have to find the expected revenue for the day if it's sunny today.
There is an 80% chance of sunshine tomorrow, which means that there is (a) 20% chance of rain or cloudy weather.
Based on this, Expected Revenue for the day
= 80% of $2,000 + 20% of ($1,600 + $400)
= $1,920
(b) Calculation of the average number of days with non-sunny weather:
Let x be the number of days the weather is not sunny.
Using the probabilities given in the question, the Probability that tomorrow will be sunny if it is sunny today = 80%
The probability that tomorrow will be rainy if it is cloudy today = 20%
The probability that tomorrow will be sunny if it is cloudy today = 30%
The probability that tomorrow will be rainy if it is cloudy today = 20%
The probability that the weather will not be sunny tomorrow if it's rainy today = 80%
Therefore, the Probability of non-sunny weather in a day,
P(x) is given as follows:
P(0) = 0.8 (When it's sunny today, there's an 80% chance it'll be sunny tomorrow)
P(1) = 0.2 × 0.3 + 0.8 × 0.2
= 0.26 (When it's cloudy today, there's a 30% chance it'll be sunny tomorrow and a 20% chance it'll be rainy)
P(2) = 0.2 × 0.7 × 0.1 + 0.8 × 0.2 × 0.2 + 0.2 × 0.3 × 0.2 + 0.8 × 0.8 × 0.1
= 0.204 (When it's rainy today, there's a 10% chance it'll be sunny tomorrow and an 80% chance it'll be rainy tomorrow).
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(x+3)(x-3)(3x+2) can you work this out please
Answer: 3x^3 + 2x^2 −27x −18
Step-by-step explanation below.
(x+3)(x−3)(3x+2)
=((x+3)(x−3))(3x+2)
=((x+3)(x−3))(3x)+((x+3)(x−3))(2)
=3x3−27x+2x2−18
=3x3+2x2−27x−18
Hope this helps!
can someone answer this pls
Answer:
5.9
Step-by-step explanation:
Answer:
I think it is 62.
Step-by-step explanation:
Hope this helps!
Martin recorded the low temperatures at his house for one week. the temperatures are shown below -
-7, -3, 4, 1, -2, -8. 7
Approximately what was the average low temperature of the week?
Answer:
The average temp was 4
Step-by-step explanation:
Consider the problem min 22 – (21 – 2)3 + 3 subject to X2 > 1 Which one is the extremizer point? O x=[0,2] O x=(2,2] O x=(2,1] Ox=[1,2] O x=(1,1] O x=[0,1] x=[2,2]
The extremizer point is x=(1,1].
Consider the problem min 22 – (21 – 2)3 + 3 subject to X2 > 1. The given problem can be written in the following form: minimize f(x)= 22 - (21 - 2)3 + 3 subject to g(x) = x2 > 1.The extremizer point is the point that satisfies the conditions that have been given in the question. Therefore, the extremizer point in this problem is x=(1,1].In order to find the extremizer point of a problem, we need to calculate the derivative of the objective function and then equate it to zero to find the point where it attains its minimum value. Here, however, we have only one point to choose from that satisfies the given constraint, so it is the extremizer point.
A point in a convex set that does not lie on any open line segment connecting two points in the set is referred to as an extreme point in mathematics.
The terms "extreme point" and "extremal point" can also mean:
a stage at which a function reaches its maximumA graph theory tree's leaf vertexExtreme points on Earth, or parts of a geographical mass that stretch farther in one direction than in any other.Know more about extremizer point here:
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what is equivilent to 0.5y + 10.5 + 6.5y - 0.5 - 0.5y
Answer:
6.5y + 10
Step-by-step explanation:
0.5y + 10.5 + 6.5y - 0.5 - 0.5y
combine like terms
6.5y + 10
Answer:
m
Step-by-step explanation:
f
plzzz helllp i need it done in 20 mins
Answer: C
Step-by-step explanation: calculate the decimal and put it into the calculator to get the fraction
Step-by-step explanation:
[tex] {10}^{ - 3} \\ = \frac{1}{10 \times 10 \times 10} \\ = \frac{1}{1000} \: or \: 0.001[/tex]
#CMIIWThe ratio of people at a baseball game to bags of peanuts sold is 12:1. If there are 480 people, how many bags of peanuts will be sold?
I don't know how you 50th your mind to the fact 7days 7days I know that was 50th and I had no doubt about that can happen in a few months but his dream is the next one to get money to go some other time and do a few more pains in your car and get a better feel of what is the best thing for the use sharks live to the point in which 50th the world is not healthy and it
Answer: 40
Step-by-step explanation: For every 12 people, 1 bag of peanuts is sold. So for 480 people, which is 40 of 12 people, there must be 40 of 1 bag of peanuts, or 40 bags of peanuts, hope this helps!
the sum of 2 numbers is -48 and their difference is 17.
Answer: -15.5, -32.5
Step-by-step explanation: Call the two numbers x and y, x+y=-48 and x-y=17, subtract both of them and you get 2y=31 and then divide both sides by 2 and get -15.5 for y, plug this back in the equations and get -32.5 for x, hope this helped!
In this problem, y = c₁e + c₂e initial conditions. y(1) = 0, y'(1) = e -x-1 y = e X s a two-parameter family of solutions of the second-order DE y" - y = 0. Find a solution of the second-order IVP consisting of this differential equation and the given
The solution to the second-order IVP consisting of the differential equation y" - y = 0 and the initial conditions y(1) = 0, y'(1) = e^(-1).
To find a solution of the second-order initial value problem (IVP) consisting of the differential equation y" - y = 0 and the given initial conditions y(1) = 0, y'(1) = e -x-1, we can follow these steps:
Determine the general solution of the differential equation y" - y = 0:
The characteristic equation is r^2 - 1 = 0. Solving this equation, we find two distinct roots: r = 1 and r = -1.
Therefore, the general solution is y(x) = c₁e^x + c₂e^(-x), where c₁ and c₂ are constants.
Apply the initial condition y(1) = 0:
Substituting x = 1 and y = 0 into the general solution:
0 = c₁e^1 + c₂e^(-1)
Dividing through by e:
0 = c₁ + c₂e^(-2)
Apply the initial condition y'(1) = e -x-1:
Differentiating the general solution:
y'(x) = c₁e^x - c₂e^(-x)
Substituting x = 1 and y' = e^(-1) into the differentiated solution:
e^(-1) = c₁e^1 - c₂e^(-1)
Dividing through by e:
e^(-2) = c₁ - c₂e^(-2)
We now have a system of two equations:
Equation 1: 0 = c₁ + c₂e^(-2)
Equation 2: e^(-2) = c₁ - c₂e^(-2)
Solving this system of equations, we can find the values of c₁ and c₂:
Adding Equation 1 and Equation 2:
0 + e^(-2) = c₁ + c₁ - c₂e^(-2)
e^(-2) = 2c₁ - c₂e^(-2)
Rearranging this equation:
2c₁ = e^(-2)(1 + c₂)
Substituting this value back into Equation 1:
0 = e^(-2)(1 + c₂) + c₂e^(-2)
0 = e^(-2) + c₂e^(-2) + c₂e^(-2)
0 = e^(-2) + 2c₂e^(-2)
-1 = 2c₂e^(-2)
Simplifying:
c₂e^(-2) = -1/2
Substituting this value back into Equation 1:
0 = c₁ - 1/2
c₁ = 1/2
Therefore, the values of c₁ and c₂ are c₁ = 1/2 and c₂ = -1/(2e^2).
Now we can write the particular solution to the IVP:
y(x) = (1/2)e^x - (1/(2e^2))e^(-x)
This is the solution to the second-order IVP consisting of the differential equation y" - y = 0 and the initial conditions y(1) = 0, y'(1) = e^(-1).
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after preparing a horizontal analysis of Blanchet
corporations balance sheet for 2019, what are some observations you
can make?
Overall, the horizontal analysis of Blanchet Corporation's balance sheet for 2019 suggests positive growth and financial stability.
There has been a significant increase in total assets from the previous year. This indicates potential growth and expansion in the company's operations.
The liabilities have also increased, but not at the same rate as the assets. This suggests that the company may have taken on additional debt or financing to support its growth.
The retained earnings have shown a positive trend, indicating that the company has been profitable and able to retain a portion of its earnings for reinvestment or future use.
The equity section has also grown, which can be attributed to the increase in retained earnings and potentially additional capital investments.
The increase in assets and retained earnings indicates the company's ability to generate profits and reinvest in its operations. However, the increase in liabilities should be carefully monitored to ensure it is manageable and sustainable in the long term.
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Help pleaseeeeeeeeeeeeeeeeeee
Is it true that for every natural number n, the integer n3 + n2 + 41 is prime? Prove or give a counterexample.
Counterexample: The statement is not true. For n = 41, the expression n^3 + n^2 + 41 equals 41^3 + 41^2 + 41, which is divisible by 41 and therefore not prime.
To prove or disprove the statement, we need to find a counterexample, i.e., a natural number n for which n^3 + n^2 + 41 is not prime. By substituting n = 41 into the expression, we obtain 41^3 + 41^2 + 41. This expression is divisible by 41 since it can be factored as 41(41^2 + 41 + 1). Since a prime number is only divisible by 1 and itself, this means that the expression is not prime and thus disproves the statement. Therefore, the claim that n^3 + n^2 + 41 is prime for every natural number n is false.
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Avery signed up for a streaming music service where there's a fixed cost for monthly
membership and a cost per song downloaded. Her total cost is given by the linear
graph below. What is the meaning of the point (1,9.24)?
Answer:the answer might be C or B
Step-by-step explanation:
The given ordered pair (1,9.24) represents the ordered pair represents the monthly cost of one song. Option C is correct.
Given that,
Avery signed up for a streaming music service where there's a fixed cost for a monthly membership and a cost per song downloaded. Her total cost is given by the linear graph.
The graph is a demonstration of curves that gives the relationship between the x and y-axis.
Here,
According to the question,
9.24 is the total cost marked on the y axis so corresponding to this the value of x is 1 which represents the number of song,
So the given ordered pair describe that the cost of one song per month is $9.24.
Thus, the given ordered pair (1,9.24) represents the ordered pair represents the monthly cost of one song. Option C is correct.
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Draw the directed graphs & zero-one matrices for each of the following relations:
a. Define a relation R on A = {0, 1, 2, 3} by R = {(0, 0), (1, 2), (2, 2)}.
b. Define a relation S on B = {a, b, c, d} by S = {(a, b), (a, c), (b, c), (d, d)}.
c. Define a relation R on A = {0, 1, 2, 3}, B= {4,5,6,8} by R = {(0, 4), (0, 6), (1, 8), (2,4), (2,5), (2,8), (3,4), (3,6)}.
In part (a), the relation R on A = {0, 1, 2, 3} is defined as R = {(0, 0), (1, 2), (2, 2)}. In part (b), the relation S on B = {a, b, c, d} is defined as S = {(a, b), (a, c), (b, c), (d, d)}. Lastly, in part (c), the relation R on A = {0, 1, 2, 3} and B = {4, 5, 6, 8} is defined as R = {(0, 4), (0, 6), (1, 8), (2, 4), (2, 5), (2, 8), (3, 4), (3, 6)}.
a) The directed graph for relation R on A = {0, 1, 2, 3} can be represented as follows:
0 -> 0
1 -> 2
2 -> 2
Here, each element of A is represented as a node, and the directed edges indicate the pairs in the relation R.
The zero-one matrix for relation R can be written as:
| 1 0 0 0 |
| 0 0 1 0 |
| 0 0 1 0 |
| 0 0 0 0 |
In this matrix, a value of 1 indicates that the corresponding pair is in the relation, and 0 indicates that it is not.
b) The directed graph for relation S on B = {a, b, c, d} is as follows:
a -> b
a -> c
b -> c
d -> d
The zero-one matrix for relation S can be written as:
| 0 1 1 0 |
| 0 0 1 0 |
| 0 0 0 0 |
| 0 0 0 1 |
c) The directed graph for relation R on A = {0, 1, 2, 3} and B = {4, 5, 6, 8} is represented as follows:
0 -> 4
0 -> 6
1 -> 8
2 -> 4
2 -> 5
2 -> 8
3 -> 4
3 -> 6
The zero-one matrix for relation R can be written as:
| 0 0 0 0 |
| 0 0 0 0 |
| 1 0 0 0 |
| 0 0 0 0 |
| 1 0 0 0 |
| 0 0 0 0 |
| 1 0 0 0 |
| 0 0 0 0 |
In this matrix, each row represents an element of A, and each column represents an element of B. A value of 1 in the matrix indicates that the corresponding pair is in the relation, and 0 indicates that it is not.
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What is the rate of change of the function Y = -1/2x + 5
Answer:
Rate of change is -1/2.
Hope this helps! :)
Answer: -1/2
Step-by-step explanation: For a function y=mx+b, m=slope, and b=y-intercept:
y=-1/2x plus 5. Rate of change = -1/2
A simple random sample of 50 items resulted in a sample mean of 25. The population standard deviation is
σ = 6.
(Round your answers to two decimal places.)
(a)
What is the standard error of the mean,
σx?
(b)
At 95% confidence, what is the margin of error?
a)The standard error of the mean is 0.85.b) the margin of error is 1.67 at 95% confidence level.
a) To calculate the standard error of the mean, the formula is given by:σx = σ / √nWhere,σ is the population standard deviation, n is the sample size√n = √50 = 7.071σx = σ / √n= 6 / 7.071σx = 0.85 (rounded to two decimal places)
b) At 95% confidence level, the margin of error is calculated by using the following formula:
ME = z* σxWhere,z* is the z-value for 95% confidence level and σx is the standard error of the mean.
The z-value for 95% confidence level is 1.96
ME = z* σx = 1.96 x 0.85
ME = 1.67 (rounded to two decimal places)
Therefore, the margin of error is 1.67 at 95% confidence level.
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Felicia is solving the expression - 5(3x + 2) + 10x. Gregory is solving the expression -12 - 4x +?+?x.
Both students simplified their answers to get the same expression. What are the values
missing from Gregory's expression?
Answer:
-5x-10 is the expression. the missing values would be 2 and -x i believe
Suppose that when your friend was born, your friend's parents deposited $4000 in an account paying 6.3% interest compounded quarterly. What will the account balance be after 13 years
Answer:
$9015.20
Step-by-step explanation:
A = 4000[1 + (.063/4)]^13·4
Answer:
3276
Step-by-step explanation:
4000(6.3*13)÷100
15 PT PLZ PLZ PLZ PLZ PLZ HELPPPPPPPPPPPPPPPPPP
Answer:
The total distance the students ran in the race would be 5 1/2 km.
Step-by-step explanation:
1/4 + 1/4 = 2/4 3/41 1/4 + 1 1/4 = 2 2/4 1 3/42/4 + 3/4 + 2 2/4 + 1 3/4 = 3 10/4 = 5 2/4 = 5 1/2 (Simplified)
Select the correct answer.
Which financial item transfers risk from one party to another?
equities or stocks
apartments
initial public offerings (IPOS)
stock futures
Submit
Answer:
Equities or stocks.
Step-by-step explanation:
Equities or stocks are the source of finance for a company. A company issues stocks and then the fund received from these stocks is used to run the business operations. Risk transfer is a way to minimize the self risk and transfer it to other person. When stocks are sold the risk is transferred to other person who buys the stocks.
Answer:
stock futures
Step-by-step explanation:
Find the radius of convergence and interval of convergence of the series. 00 2. ενη -(x+6) " 8n n=1
The series converges for -8 < x < -4, and the radius of convergence is 2.
To find the radius of convergence and interval of convergence of the series, we'll use the ratio test.
The series given is:
Σ (n=1 to ∞) [tex]2^{(-n)}(x + 6)^n[/tex]
We'll apply the ratio test to determine the convergence behavior:
lim(n→∞) |(a_{(n+1)})/(a_n)|
= lim(n→∞) |[tex][2^{(-(n+1)})(x + 6)^{(n+1)}] / [2^{(-n)}(x + 6)^n][/tex]|
= lim(n→∞) |[tex][2^{(-(n+1)})(x + 6)] / [2^{(-n)}][/tex]|
= lim(n→∞) |(x + 6)/2|
To ensure convergence, we need the above limit to be less than 1:
|(x + 6)/2| < 1
Now, let's consider the cases when the above inequality holds true:
Case 1: (x + 6)/2 < 1
Simplifying, we get:
x + 6 < 2
x < -4
Case 2: -(x + 6)/2 < 1
Simplifying, we get:
x + 6 > -2
x > -8
Combining the results from both cases, we have:
-8 < x < -4
Therefore, the interval of convergence is -8 < x < -4.
To find the radius of convergence, we consider the endpoints of the interval of convergence (-8 and -4). The radius of convergence (R) is half the length of the interval of convergence.
R = (|-8 - (-4)|)/2
= 4/2
= 2
Hence, the radius of convergence is 2.
In summary, the series converges for -8 < x < -4, and the radius of convergence is 2.
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I need help im struggling plz help
Answer:
I believe the circumference is 37.699
HOPE I HELPED!
PLEASE GIVE BRAINLIEST!
Step-by-step explanation:
Answer:
37.68 cm
Step-by-step explanation:
The photo shows the diameter as 12cm
The formula for the circumference of a circle is [tex]2\pi r[/tex]
If the diameter is 12, the radius is 6
[tex]2 \pi 6[/tex] = 2 x 3.14 x 6 = 37.68
Find a power series representation for the function. +3 f(x) = (x – 4)2 00 f(x) = Ï ) n = 0 Determine the radius of convergence, R. R=
The power series representation of f(x) = (x - 4)² is f(x) = 9 - 6x + x² + ..., and the radius of convergence, R, is infinity (∞).
To find a power series representation for the function f(x) = (x - 4)², we can expand it using the binomial theorem.
The binomial theorem states that for any real number r and a real number x such that |x| < 1, we have:
[tex](1 + x)^r[/tex] = 1 + rx + (r(r-1))/2! * x² + (r(r-1)(r-2))/3! * x³ + ...
In our case, we have f(x) = (x - 4)², which can be rewritten as (1 + (x - 4))². Using the binomial theorem with r = 2, we get:
f(x) = (1 + (x - 4))² = 1 + 2(x - 4) + (2(2-1))/2! * (x - 4)² + ...
Simplifying this expression, we have:
f(x) = 1 + 2x - 8 + (2/2) * (x² - 8x + 16) + ...
Expanding further, we get:
f(x) = 1 + 2x - 8 + x² - 8x + 16 + ...
Now we can write the power series representation of f(x) as:
f(x) = 9 - 6x + x² + ...
To determine the radius of convergence, R, we need to find the interval of x for which the series converges. In this case, the series converges for all real numbers x since there are no terms involving powers of x that could cause divergence.
Therefore, the radius of convergence, R, is infinity (∞).
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Solve the equation 6x –3y = 9 for y
Answer:
y= -3
Step-by-step explanation:
Substitute x as 0
6(0)-3y=9
-3y=9
y= -3
Not a question but need help. Check out the image.
Answer:
well you need to have your friends personal email then login to you email and press new message once your logged in put in you friends email and you are all set
Step-by-step explanation:
HELPPPP PLSSS I WILL MARK BRAINLYEST
Answer:
B.
Step-by-step explanation:
hope this helps!!
Let X and Y be independently random variables, with X uniformly distributed on [0, 1] and Y uniformly distributed on (0,2). Find the PDF fz (z) of Z = max{X,Y}.
1. For z < 0 or z > 2: fz (z) =
2. For 0
3. For 1
The probability density function (PDF) of the random variable Z, defined as the maximum of X and Y, can be determined as follows:
For z < 0 or z > 2: fz(z) = 0
For 0 < z < 1: fz(z) = z
For 1 < z < 2: fz(z) = 2 - z
To find the PDF of Z, we need to consider the different regions of the interval [0, 2] and determine the probability density function for each region.
For z < 0 or z > 2:
Since Z cannot be less than 0 or greater than 2, the probability density in these regions is 0. Therefore, fz(z) = 0 for z < 0 or z > 2.
For 0 < z < 1:
In this range, the maximum of X and Y is always Y because Y ranges from 0 to 2. Therefore, we can write fz(z) = P(Z < z) = P(Y < z). Since Y is uniformly distributed on (0, 2), its PDF is constant within this range. The probability of Y being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 2), which is z/2. Therefore, fz(z) = z/2 for 0 < z < 1.
For 1 < z < 2:
In this range, the maximum of X and Y can either be X or Y. To determine the PDF, we need to consider the probability of X being the maximum and the probability of Y being the maximum.
Probability of X being the maximum:
Since X is uniformly distributed on [0, 1], the probability of X being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 1), which is z/1 = z. Therefore, the probability of X being the maximum is z.
Probability of Y being the maximum:
Since Y is uniformly distributed on (0, 2), the probability of Y being less than z is given by the ratio of the length of the interval (0, z) to the length of the interval (0, 2), which is z/2. Therefore, the probability of Y being the maximum is z/2.
Since X and Y are independent, we can add their probabilities to find the overall probability of Z being less than z. Therefore, fz(z) = P(Z < z) = P(X < z) + P(Y < z) = z + z/2 = 2z/2 + z/2 = (3z)/2.
To find the PDF fz(z) in this range, we need to calculate the derivative of the cumulative distribution function (CDF). The CDF of Z can be obtained by integrating the PDF fz(z):
Fz(z) = ∫[0,z] fz(u) du
Taking the derivative of Fz(z) with respect to z, we get:
fz(z) = d/dz (Fz(z)) = d/dz ∫[0,z] fz(u) du = d/dz (3z^2/4) = 3z/2.
Therefore, fz(z) = 3z/2 for 1 < z < 2.
For z < 0 or z > 2: fz(z) = 0
For 0 < z < 1: fz(z
To learn more about the cumulative distribution function
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