a. To determine if the two populations of crawfish from Devon and Cornwall have the same mean carapace length, we can use the two-sample t-test.
This test compares the means of two independent samples to assess whether they are significantly different.
We can calculate the sample means and sample standard deviations for both groups:
Next, we calculate the t-value using the formula:
To determine if this t-value is statistically significant, we need to compare it to the critical value from the t-distribution for the given degrees of freedom
If the calculated t-value falls outside the critical region, we can reject the null hypothesis and conclude that the two populations have different mean carapace lengths. Otherwise, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in the mean carapace length between the two populations.
b. To answer the question using the Wilcoxon rank sum test, we need to combine the observations from both samples, assign ranks based on their relative positions, and calculate the sum of ranks for one of the samples (either Devon or Cornwall). The null hypothesis for this test is that the two distributions are the same.
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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)
Find the interval [μ - z (σ/sqrt(n)), μ + z (σ/sqrt(n))] within which 95 percent of the sample means would be expected to fall, assuming that each sample is from a normal population.
a. μ = 173, σ = 20, n = 42.
b. μ = 874, σ = 12, n = 7.
c. μ = 76, σ = 2, n = 26.
The interval [μ - z (σ/sqrt(n)), μ + z (σ/sqrt(n))] within which 95 percent of the sample means would be expected to fall, assuming that each sample is from a normal population.
a. The interval is [166.01, 179.99].
b. The interval is [849.07, 898.93].
c. The interval is [74.47, 77.53].
a. μ = 173, σ = 20, n = 42.
Here, we have, μ = 173, σ = 20, n = 42.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 173 - 1.96(20/sqrt(42)) , 173 + 1.96(20/sqrt(42)) ]
i.e [ 166.01, 179.99 ]
Therefore, the interval is [166.01, 179.99].
b. μ = 874, σ = 12, n = 7.
Here, we have, μ = 874, σ = 12, n = 7.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 874 - 1.96(12/sqrt(7)) , 874 + 1.96(12/sqrt(7)) ]
i.e [ 849.07, 898.93 ]
Therefore, the interval is [849.07, 898.93].
c. μ = 76, σ = 2, n = 26.
Here, we have, μ = 76, σ = 2, n = 26.
Using the z-table, we get z = 1.96 (at 95% confidence level).
The interval is: [ 76 - 1.96(2/sqrt(26)) , 76 + 1.96(2/sqrt(26)) ].
i.e [ 74.47, 77.53 ]
Therefore, the interval is [74.47, 77.53].
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HELPPPP PLSSS I WILL MARK BRAINLYEST
Answer:
B.
Step-by-step explanation:
hope this helps!!
X/8 < -1 please help
Answer:
x<-8
Step-by-step explanation:
x/8<-1
x<-1×8
x<-8
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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Help pleaseeeeeeeeeeeeeeeeeee
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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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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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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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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Pls help, question on picture, will do brainliest if right
no links!!!!!
Answer: 24/25
Step-by-step explanation:
Sin = opposite/hypotenuse
Sin = 24/25
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:
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
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 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
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!
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
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:
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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What is the purpose of scientific notation? How is scientific notation represented? Explain.
plz help
Answer:
Step-by-step explanation: The purpose of scientific notation is to make the numbers and quantities used easier to comprehend, to read and to write.
Express both numbers with the same power of ten.
Add the base numbers.
Bring the power of ten down to represent the new power of ten for the sum.
Simplify so that the base number is between 1 and 10.
Answer:
What is the purpose of scientific notation?
"to make the numbers and quantities used easier to comprehend, to read and to write. " <----- Credits to: Yahoo
How is scientific notation represented?
"To start with, scientific notation is a form of expressing very small or large numbers in a simpler form." <--------- Credits to: Yahoo
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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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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Given an investment of $1,500: which investment would have a larger balance after 5 years? Option 1 - 4% compounded monthly option 2 - 3.9% compounded daily.
Answer:
It is option one
Step-by-step explanation:
I don’t know why but I’m doing a quizzizz with the same question and i picked option 2 but option one was the correct answer
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
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
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
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:
PLEASE HELP I DONT UNDERSTAND I WILL GIVE EXTRA POINTS IF ITS RIGHT!!!!!!!!!!!!
Answer:
10.7
Step-by-step explanation:
Hello There!
The image shown below shows the relationship between two chords formed inside of a circle
So the product of the lengths of the lines in the same chord is equal to the product of the other length of the chords lines if that makes sense
So basically 6 * 16 = 9 * x
we can now use this equation that was just made to solve for x
6 * 16 = 96
96 = 9x
*divide each side by 9*
96/9=10.66666667 *round to the nearest tenth* 10.7
9x/x
we're left with x = 10.7
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:
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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