Answers
a) If f(y) is a probability density function, then both f(y) ≥ 0 for all y in its support, and the integral of f(y) over its entire support should be 1. eˣ > 0 for all real x, so the first condition is met. We have
[tex]\displaystyle \int_{-\infty}^\infty f(y) \, dy = \frac14 \int_0^\infty e^{-\frac y4} \, dy = -\left(\lim_{y\to\infty}e^{-\frac y4} - e^0\right) = \boxed{1}[/tex]
so both conditions are met and f(y) is indeed a PDF.
b) The probability P(Y > 4) is given by the integral,
[tex]\displaystyle \int_{-\infty}^4 f(y) \, dy = \frac14 \int_0^4 e^{-\frac y4} \, dy = -\left(e^{-1} - e^0\right) = \frac{e - 1}{e} \approx \boxed{0.632}[/tex]
c) The mean is given by the integral,
[tex]\displaystyle \int_{-\infty}^\infty y f(y) \, dy = \frac14 \int_0^\infty y e^{-\frac y4} \, dy[/tex]
Integrate by parts, with
[tex]u = y \implies du = dy[/tex]
[tex]dv = e^{-\frac y4} \, dy \implies v = -4 e^{-\frac y4}[/tex]
Then
[tex]\displaystyle \int_{-\infty}^\infty y f(y) \, dy = \frac14 \left(\left(\lim_{y\to\infty}\left(-4y e^{-\frac y4}\right) - \left(-4\cdot0\cdot e^0\right)\right) + 4 \int_0^\infty e^{-\frac y4} \, dy\right)[/tex]
[tex]\displaystyle \cdots = \int_0^\infty e^{-\frac y4} \, dy[/tex]
[tex]\displaystyle \cdots = -4 \left(\lim_{y\to\infty} e^{-\frac y4} - e^0\right) = \boxed{4}[/tex]
Related Questions
Solve using substitution. (Explanation too please)
y = 7x + 2
y = 10x − 1
Answers
Michael is saving money to buy a bike. So far, he
has saved $25 which is four-fifths of the total cost
of the bike. How much does the bike cost?
Answers
Let X = total cost of game
(4/5) X = $25
Multiply both sides of equation by 5/3
(5/3)(3/5) X = (5/3) $25
X = $35
What is 10% of 75? Use the double number line below to label your work and answer.
Answers
Answer: the answer is 7.5
Step-by-step explanation:
please helppppppppppppp ASAP. please ensure to use step by step explanation. thanks.
Answers
I'll do the first two parts to get you started.
=====================================================
Part (i)
We can show that the operation [tex]\nabla[/tex] is not commutative by picking two random and different real numbers for a,b. In other words, I'm using a counter-example.
Let's say we go for a = 1 and b = 2
[tex]a \nabla b = \frac{3a+b}{5} - 1\\\\1 \nabla 2 = \frac{3*1+2}{5} - 1\\\\1 \nabla 2 = \frac{5}{5} - 1\\\\1 \nabla 2 = 1 - 1\\\\1 \nabla 2 = 0\\\\[/tex]
Now let's swap the values. We'll try a = 2 and b = 1
[tex]a \nabla b = \frac{3a+b}{5} - 1\\\\2 \nabla 1 = \frac{3*2+1}{5} - 1\\\\2 \nabla 1 = \frac{7}{5} - 1\\\\2 \nabla 1 = 1.4 - 1\\\\2 \nabla 1 = 0.4\\\\[/tex]
We can see that [tex]1 \nabla 2[/tex] and [tex]2 \nabla 1[/tex] are not the same value. In general, [tex]a \nabla b \ne b \nabla a[/tex] Therefore, the operation [tex]\nabla[/tex] is not commutative.
The only time [tex]a \nabla b = b \nabla a\\\\[/tex] is true is when [tex]a = b[/tex], since,
[tex]\frac{3a+b}{5}-1 = \frac{3b+a}{5}-1\\\\\frac{3a+a}{5}-1 = \frac{3a+a}{5}-1\\\\\frac{4a}{5}-1 = \frac{4a}{5}-1\\\\[/tex]
It's when [tex]a \ne b[/tex] is where the operation becomes noncommutative.
Another way to arrive at the [tex]a = b[/tex] condition is to solve the original equation for either 'a' or b.
=====================================================
Part (ii)
Using the previous part as inspiration, we'll do a counter-example to show that the operation is not associative. Pick 3 random values for a,b,c. Here are the values I'll pick.
a = 1b = 2c = 3Then,
[tex]d = b \nabla c = \frac{3b+c}{5}-1 = \frac{3*2+3}{5}-1 = 0.8\\\\a \nabla (b \nabla c) = a \nabla d = \frac{3a+d}{5}-1 = \frac{3*1+0.8}{5}-1 = -0.24\\[/tex]
Next, we can say:
[tex]d = a \nabla b = \frac{3a+b}{5}-1 = \frac{3*1+2}{5}-1 = 0\\\\(a \nabla b) \nabla c = d \ \nabla c = \frac{3d+c}{5}-1 = \frac{3*0+3}{5}-1 = -0.4[/tex]
Let's compare the two outputs:
[tex]a \nabla (b \nabla c) = -0.24\\\\(a \nabla b) \nabla c = -0.4[/tex]
They don't match up, so [tex]a \nabla (b \nabla c) \ne (a \nabla b) \nabla c[/tex] when (a,b,c) = (1,2,3).
In general, the operation is not associative in R.
What are the coordinates of the point (-4, 2) after a translation 2 units left and 2 units up?
A (-2, 0)
B (-6,0)
C (-2, 4)
D (-6, 4)
Answers
Answer:
the answer you're looking for is D. when you move to the left the -4 becomes -6 and when you move up 2 becomes 4.
what do i do for the 2 boxes circled in purple? help!
Answers
Which of the following best describes the slope of the line below?
Answers
Answer:
Negative
Step-by-step explanation:
The slope of the line is negative because when looking at the graph, you can see that the line starts in the 2nd quadrant and ends in the 4th.
Need help on True or False PLEASE answer!!!!
Answers
Answer:
Step-by-step explanation:
1. False
2. False
3. True
4. True
5. True
6. True
7. True
8. True
9. True
Expressions equivalent to 8 to the negative 2 power times 3 to the negative 2 power
Answers
Answer:
0.0007361111111111111+0.001;
0.0027361111111111111-0.001;
1.0017361111111111111-1;
Step-by-step explanation:
you calculate 8^(-2)*3^(-2) that gives you 0.0017361111111111111 and then you add 1 anywhere you want and then say -1 where you put that +1 and that's all
what is /12x^8| /3x2 in simplest form where x>0
Answers
Answer:
2x^3
C
Step-by-step explanation:
i need help plz
thanks
Answers
Troy is buying a car that costs $15,000. He
plans to get a 5-year loan to pay for it. He
can get a loan for $15,000 or he can pay
$3,000 from his savings and get a loan
for the rest. The savings account pays 2%
simple interest per year. The simple interest
rate for the loan is 0.5% per year.
a. How much interest over a 5-year
Answers
Answer:
300/75
Step-by-step explanation:
Cost of the Car = $15,000
a) Rate of interest received on savings(r) = 2%
Amount in savings account(p) =$3000
Time period(t) = 5 years
simple intrest.
Interest received on savings account = $300
b) Rate of interest on loan = 0.5%
IF P= 15000
intrest.
Difference between the interest = $75
c) Taking loan of the whole amount that is of $15,000 is more reasonable because though the interest is more but Troy will receive interest($300) from his savings account as well. But if he withdraws $3000 from savings and takes the loan for the rest of the amount, he would have no earnings.
The radioactive compound 14CO2 may be used as a tracer to study metabolism in plants. Suppose that a compound isolated from a plant exhibited 28, 32, 27, 39 and 40 counts of radioactive decays per minute. A blank sample used to measure the background counts of the radiation counter gave 28,21, 28 and 20 counts per minute. It appears that the isolated compound gives more counts than those from background. Can we be 95% confident that the compound is indeed radioactive?
Answers
Using the t-distribution, it is found that since the test statistic is more than the critical value for the right-tailed test, it can be conclude that the isolated compound gives more counts than those from background, and hence, we can be 95% confident that the compound is indeed radioactive.
At the null hypothesis, it is tested if isolated compounds do not give more counts than those from background, that is:
[tex]H_0: \mu_1 - \mu_2 \leq 0[/tex]
At the alternative hypothesis, it is tested if the give more counts, that is:
[tex]H_1: \mu_1 - \mu_2 > 0[/tex]
Using a calculator, the mean and the standard deviation for each sample are given by:
[tex]\mu_1 = 33.2, s_1 = 6.058[/tex]
[tex]\mu_2 = 24.25, s_2 = 4.35[/tex]
Considering the sample sizes of 5 and 4, respectively, the standard errors are given by:
[tex]s_1 = \frac{6.058}{\sqrt{5}} = 2.71[/tex]
[tex]s_2 = \frac{4.35}{\sqrt{4}} = 2.175[/tex]
For the distribution of differences, the mean and standard error are given by:
[tex]\overline{x} = \mu_1 - \mu_2 = 33.2 - 24.25 = 8.95[/tex]
[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{2.71^2 + 2.175^2} = 3.4749[/tex]
The standard error was found from the standard deviation for each sample, hence, the t-distribution is used.
The test statistic is given by:
[tex]t = \frac{\overline{x} - \mu}{s}[/tex]
In which [tex]\mu = 0[/tex] is the value tested at the null hypothesis, hence:
[tex]t = \frac{\overline{x} - \mu}{s}[/tex]
[tex]t = \frac{8.95 - 0}{3.4749}[/tex]
[tex]t = 2.58[/tex]
The critical value for a right-tailed test, as we are testing if the mean is greater than a value, with 5 + 4 - 2 = 7 df and a 0.05 significance level is of [tex]t^{\ast} = 1.895[/tex]
Since the test statistic is more than the critical value for the right-tailed test, it can be conclude that the isolated compound gives more counts than those from background, and hence, we can be 95% confident that the compound is indeed radioactive.
A similar problem is given at brainly.com/question/24826023
What does Point P on the number line represent? (Use the hyphen for negative numbers, such as −9)
Number line from negative 8 to positive 10 in increments of 1 is shown. Only the even numbers are labeled. A point labeled P is placed at the third tick mark to the left of 0.
(5 points)
Answers
Answer:
P = -3, each line in the negative direction is 1 number
so you would have -1, -2, -3 , -4, etc.
P is on the -3 line.
See attached picture.
Step-by-step explanation:
A scuba diver descends farther down into the ocean from an initial depth of 12.8 feet below sea level. The scuba diver descends at a constant rate for 312 minutes and reaches a depth no more than 49.2 feet below sea level.
What is the maximum rate, r, that the scuba diver may descend?
r≥10.4
r≤10.4
r≥13 7/9
r≤13 7/9
Answers
Answer:
[tex]r \geqslant 10.4[/tex]
The maximum rate at which the diver will descend will be 7 feet per hour.
What is the rate of change?The rate of change is the change of a quantity over 1 unit of another quantity.
Most of the time the rate of change is the change with respect to time.
For example the speed 3meter/second.
As per the given,
Initial depth of scuba diver = 12.8 feet
Let's say the rate of a scuba diver is "r".
Initial depth + depth cover with constant rate ≤ final depth
12.8 + (312/60)r ≤ 49.2
r ≤ 7
Thus, the maximum rate at which a diver can descend is 7 feet per hour.
Hence"The maximum rate at which the diver will descend will be 7 feet per hour".
To learn more about the rate of change here,
brainly.com/question/12786410
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Please explain how you got to the answer. I am trying to understand the concept that went into it.
Solve for x in the equation x squared + 11 x + StartFraction 121 Over 4 EndFraction = StartFraction 125 Over 4 EndFraction.
A. x = negative 11 plus-or-minus StartFraction 25 Over 2 EndFraction
B. x = negative eleven-halves plus-or-minus StartFraction 25 Over 2 EndFraction
C. x = negative 11 plus-or-minus StartFraction 5 StartRoot 5 EndRoot Over 2 EndFraction
D.x = negative eleven-halves plus-or-minus StartFraction 5 StartRoot 5 EndRoot Over 2 EndFraction
Answers
Answer:
D. x = negative eleven-halves plus-or-minus StartFraction 5 StartRoot 5 EndRoot Over 2 EndFraction
Step-by-step explanation:
Evaluate 0 and 1 (xe+ex)Dx
What are the domain and range of the function below? (In picture)
Answers
Answer:
Domain: [0 , ∞)
Range: (-∞ , 4]
Step-by-step explanation:
I) DOMAIN:For domain, look along the x axis.
The graph starts from x = 0 and it's ending point isn't given in the graph as it continues moving along the positive x axis.
Hence, it's ending point must be ∞
Domain of a function is:
[minimum point in x, maximum point in x]
=> [0, ∞)
"),(" is used for infinity.
"],[" for points that lie within the domain of the function.
II) RANGEFor range, look along the y axis.
The graph starts at y = 4, and continues moving downwards till infinity.
Range:
[minimum in y, maximum in y]
=> (-∞, 4]
For parenthesis, same rule as domain will be applied. 4 is naturally greater than all values in negative. Hence, maximum point will be 4 and minimum will be -∞.Write an equation in slope-intercept form for the line with slope 6 and y-intercept -3
Answers
Answer:
y = 6x - 3
Step-by-step explanation:
Answer asap! Need help before I go to sleep.
Answers
Answer:
Step-by-step explanation:
y = (4/7)x - 1
OMG HELP RN I MARK BARINLIST
Which of the following correctly describes the solutions to the inequality?
A.
There are an infinite number of solutions. The solutions are all numbers less than 2.
B.
There are an infinite number of solutions. The solutions are all numbers greater than 2.
C.
There are 7 solutions. The solutions are the numbers between 2 and 8.
D.
There are 4 solutions. The solutions are the numbers between -2 and 1.
Answers
Answer:
B
Step-by-step explanation:
The dot on 2 is an open spot, so 2 isn't part of the solution. the arrow points towards +∞ so, yeah. b
HELP ASAP Match each division expression to its quotient.
Answers
Answer:
2=-12.2/(-6.1
-2=16/(-8)
-3= -2 2/5/4/5
3= 3 3/7 / 1 1/7
Step-by-step explanation:
Which equation describes the line shown in the graph?
Answers
Answer:
D
Step-by-step explanation:
:)
2 with the radical of 32
Answers
In Ms. Rivera’s desk, there are 14 yellow markers. There are 8 more pink markers than yellow markers, and 6 more blue than pink. How many markers does Mrs. Rivera have in her desk?
Answers
Answer:
64
Step-by-step explanation:
since ms. rivera has 8 more pink markers than yellow markers you would add 8 and get 22, so she has 22 pink markers and 14 yellow. for the blue markers it says she has 6 more blue markets than pink so you would add 6 to the 22 pink markers and get 28.
So ms. rivera has 14 yellow, 22 pink, 28 blue
add them all together and you get 64.
The total number of markers Mrs. Rivera have in her desk is 64.
Given that, Ms. Rivera’s desk, there are 14 yellow markers.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Let the total number of markers be x.
There are 8 more pink markers than yellow markers
So, number of pink markers = 14+8
= 22
There are 6 more blue markers than pink markers
So, number of blue markers = 22+6
= 28
Here, x= Number of yellow markers + Number of pink markers + Number of blue markers
= 14+22+28
= 64
Therefore, the total number of markers Mrs. Rivera have in her desk is 64.
To learn more about an equation visit:
https://brainly.com/question/14686792.
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please help me out on this one
Answers
Answer:
6/6 or 1
Step-by-step explanation:
bow tie method
What’s the difference for this equation!!!
Answers
3x^2+9x+13 – x^2+2x–12
= 2x^2 + 11x – 1
Answer:
2x^2 + 11x – 1
Step-by-step explanation:
Can anyone help with this?
Answers
The third one 3
Hope it help
128x^2-162
How do you factor this
Answers
If any doubt leave a comment
Please help with this!
Answers
Answer:
I think this is the answer
Step-by-step explanation:
p= x+2+x+9+2x
Simplified
4xin+11in
Anita earns 40 points every time she shops at Brookshire's. She needs a total of 1,500 points to receive a free set of dishes. So far she has earned 300 points. How many more times will Anita need to shop at the grocery store to earn the points she needs to earn the dishes?
a. 1800
b. 30
c. 1200
d. 45
Answers
1500 - 300 = 1200
1200/40 = 30
We know that:
1500 - 300 = required points to receive a free set of dishes1 day = 40 points.Now, let's simplify.
=> 1500 - 300 = required points to receive a free set of dishes=> 1200 = required points to receive a free set of dishes=> 1 time = 40 points=> 1 time × 30 = 40 × 30 [30 x 40 = 1200]=> 30 times = 1200 pointsConclusion:Therefore, Anita will need to visit Brookshire 30 more times to receive a free set of dishes.
Hoped this helped
[tex]BrainiacUser1357[/tex]