I'm assuming
[tex]f(y)=\begin{cases}ky(1-y)&\text{for }0\le y\le1\\0&\text{otherwise}\end{cases}[/tex]
(a) f(x) is a valid probability density function if its integral over the support is 1:
[tex]\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=k\int_0^1 y(1-y)\,\mathrm dy=k\int_0^1(y-y^2)\,\mathrm dy=1[/tex]
Compute the integral:
[tex]\displaystyle\int_0^1(y-y^2)\,\mathrm dy=\left(\frac{y^2}2-\frac{y^3}3\right)\bigg|_0^1=\frac12-\frac13=\frac16[/tex]
So we have
k / 6 = 1 → k = 6
(b) By definition of conditional probability,
P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4 and Y ≤ 0.8) / P(Y ≤ 0.8)
P(Y ≤ 0.4 | Y ≤ 0.8) = P(Y ≤ 0.4) / P(Y ≤ 0.8)
It makes sense to derive the cumulative distribution function (CDF) for the rest of the problem, since F(y) = P(Y ≤ y).
We have
[tex]\displaystyle F(y)=\int_{-\infty}^y f(t)\,\mathrm dt=\int_0^y6t(1-t)\,\mathrm dt=\begin{cases}0&\text{for }y<0\\3y^2-2y^3&\text{for }0\le y<1\\1&\text{for }y\ge1\end{cases}[/tex]
Then
P(Y ≤ 0.4) = F (0.4) = 0.352
P(Y ≤ 0.8) = F (0.8) = 0.896
and so
P(Y ≤ 0.4 | Y ≤ 0.8) = 0.352 / 0.896 ≈ 0.393
(c) The 0.95 quantile is the value φ such that
P(Y ≤ φ) = 0.95
In terms of the integral definition of the CDF, we have solve for φ such that
[tex]\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=0.95[/tex]
We have
[tex]\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=\int_0^\varphi 6y(1-y)\,\mathrm dy=(3y^2-2y^3)\bigg|_0^\varphi = 0.95[/tex]
which reduces to the cubic
3φ² - 2φ³ = 0.95
Use a calculator to solve this and find that φ ≈ 0.865.
Using probability concepts, it is found that
a) The value is k = 6.
b) P(Y ≤ .4|Y ≤ .8) = 0.554.
c) The 95th percentile is x = 0.86465.
The density function is:
[tex]f(y) = ky(1 - y)[/tex]
Item a:
The condition that makes it a probability density function is:
[tex]\int_0^1 f(y) dy = 1[/tex]
Thus:
[tex]\int_0^1 f(y) dy = \int_0^1 ky - ky^2 dy[/tex]
[tex]\int_0^1 f(y) dy = k(\frac{y^2}{2} - \frac{y^3}{3})|_{y = 0}^{y = 1}[/tex]
[tex]\int_0^1 f(y) dy = k(\frac{1}{2} - \frac{1}{3})[/tex]
Then
[tex]k(\frac{1}{2} - \frac{1}{3}) = 1[/tex]
[tex]k(\frac{3-2}{6}) = 1[/tex]
[tex]k = 6[/tex]
The value is k = 6.
Item b:
This probability is:
[tex]\int_0.4^0.8 f(y) dy = 6(\frac{y^2}{2} - \frac{y^3}{3})|_{y = 0.4}^{y = 0.8}[/tex]
Then
[tex]p = 6\left(\frac{0.8^2}{2} - \frac{0.8^3}{3} - \frac{0.4^2}{2} + \frac{0.4^3}{3}\right)[/tex]
[tex]p = 0.554[/tex]
Thus, P(Y ≤ .4|Y ≤ .8) = 0.554.
Item c:
This is x for which:
[tex]\int_0^x f(y) dy = 0.95[/tex]
Thus:
[tex]6(\frac{x^2}{2} - \frac{x^3}{3}) = 0.95[/tex]
[tex]-2x^3 + 3x^2 - 0.95 = 0[/tex]
Solving a cubic equation with the help of a calculator, and considering [tex]0 \leq x \leq 1[/tex], this is x = 0.86465.
A similar problem is given at https://brainly.com/question/22223588
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What are the elements of (A ∩ B) ' ?
Answer:
1 and 2 are the elements of (A B)
Rewrite the expression in the form a^n
Answer:
a7/5
Step-by-step explanation:
The point-slope form of the equation of a line that passes through points (8, 4) and (0, 2) is y - 4 = (x – 8). What
is the slope-intercept form of the equation for this line?
O y = 1 x - 12
O y = -x-4
O y=+*+2
O y=x+6
Answer: y = x/4 + 2
Y equals one fourth x plus two.
Step-by-step explanation:
There seems to be a typo on answer choice c.
Slope intercept form is y= mx + b
m is the slope. b is the y-intercept.
The y-intercept is given here in the coordinate (0,2) It is +2
Slope is rise/run.
Rise: Difference in given y-values is 4-2 = 2.
Run: Difference in x-values is 8-0=8.
So the slope is 2/8 = 1/4
The equation is y = 1/4 x + 2
sergio must ride 7.5 miles in 1/4 hour to achieve this target speed, so he needs to ride an additional ____ mile per 1/4 hour
Answer: 7.0
Step-by-step explanation:
sergio must ride 7.5 miles in 1/4 hour to achieve this target speed so he needs to ride an addititonal 9.8 mile per1/4 hour
Answer:
30 mph
Step-by-step explanation:
since it is 1/4 you would multply 7.5 times 4, which adds up to 30.
Write the next three terms: 7,10,13,16,
Answer:
7,10,13,16,19,21,24,27
Step-by-step explanation:
Add 3 every time
I hope this helped you! If it did, please consider rating, pressing thanks, and giving my answer 'Brainliest.' Have a great day! :)
Captain America's circular shield has a diameter of 76.2 centimetres. What is the area of his shield?
Answer:
Its area is 370 square centimeters.
Step-by-step explanation:
What is an equation in point-slope form of the line that passes through (1, −6) and (4, 6)?
Answer:
y=4x-10
Step-by-step explanation:
PLEASE HELP ASAP!?!?!?!?!!!?!?!?!?!?!?!
The diameter of a pizza is 13 1/4 inches what is the length in inches of the diameters of 6 pizzas?
Answer:
79.5
Step-by-step explanation:
A train travels 558 miles in 3 hours. At this rate, how far does the train travel
in 6 hours?
186 miles
O 1,674
0
1,116 miles
O 18 miles
Answer:
the answer is 1,674
Step-by-step explanation:
because if 3hours in 558 miles you multiply the 3hours in 558miles and the answer is 1,674
brainliest will be given if correct
Answer:
10
Step-by-step explanation:
im not sure if its correct but area is the area around the trapezoid and i added them
87.039 as a mixed number
Answer:
87 39/1000
Step-by-step explanation:
Which equation represents the following word problem? Grace is making a cake. The recipe calls for 21/2 cups of flour. If she has already added 3/4 of a cup into the bowl, how much more flour will she need to add? Let x be the number of cups she needs to add.
A) x+3/4 = 22/2
B) x-3/4=27/2
C) 3/4x = 21/2
D) X/13/4)
Answer: c
Step-by-step explanation:
3/4x = 21/2
cooper's jump rope is for 4 2/5. luke's jump rope is 4 3/4 ft long. whose jump rope is longer? how much longer? show your work
Answer:
Juke's jump; by 0.35 feet
Step-by-step explanation:
Given that,
Cooper's jump rope is for [tex]4\dfrac{2}{5}[/tex] feet.
Juke's jump rope is [tex]4\dfrac{3}{4}[/tex] feet.
Cooper's jump is :
[tex]4\dfrac{2}{5}=\dfrac{22}{5}\ \text{feet}\\\\=4.4\ \text{feet}[/tex]
Juke's jump is
[tex]4\dfrac{3}{4}=\dfrac{19}{4}\\\\=4.75\ \text{feet}[/tex]
So, it is clear that Juke's jump is longer than Cooper's jump.
Juke's jump - Cooper's jump = 4.75 - 4.4
= 0.35 feet
or
So, Juke's jump is 375 feet longer.
will give brainliest!!!! The value of the 5 in the number 594,631 is how many times larger than the value of the 5 in the number 859,463? *
Answer:
It's 10 times larger.
Step-by-step explanation:
If you just have 500,000, and the other one is 50,000, That means 50,000 x 10 is 500,000.
24% of 72 is what number? (will mark brainliest for full explanation)
Answer:
300
Step-by-step explanation:
Answer:
17.28
Step-by-step explanation:
Multiply 72 by .24, .24 being 24% because it is 24 out of 100.
What is the slope of a line passing
through points (-6, 3) and (-3, -1)?
Answer:
-4/3
Step-by-step explanation:
you do change in y over change in x which is -1-3 over -3-(-6)
when you do the calculations you get -4/3
If a bike is going at 26 centimeters per second how many meters per minute is she traveling?
Answer:
15.6 meters in a minute
Step-by-step explanation:
Here's why:
1 centimeter= 0.01 meter
So, if you have 26 cm, that equals 0.26 meters.
Now, we know that there are 60 seconds in a minute.
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I hope this helps you! :)
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Answer:
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Step-by-step explanation:
A number decreased by 22 is 44. Which equation can be used to solve this problem
Answer:
44+22=x or x-22=44
Step-by-step explanation:
X is the unknown, and so x decreased (minus) 22 is(equals) 44, so x-22=44.
Hope this helps please mark brainliest :D
Julie sent 200 emails last month. This month she sent 40% less. How many
emails did she send this month?
Answer:
She sent 120 emails this month.
Step-by-step explanation:
Multiply 200 by .60 (60 percent) and it will give you 120.
I used 60 instead of 40 because the problem said 40 percent less, NOT 40 percent of 200.
Could i get any help here?
Answer:
You will need to figure out how much it is going up by
Step-by-step explanation:
-6n + 3= 45
how do you solve this
Answer:
n= -7
Step-by-step explanation:
subtract 3 from both sides,-6n=42,divide both sides by -6
Answer:
[tex]\boxed{n = - 7}[/tex]
Step-by-step explanation:
[tex] \boxed{you \: solve \: by \: making \to n :} \\ the \: subjet \: of \: the \: equation \to \: \\ it \: means \: making \: \to \: n : to \: stand \: on \: its \: own \to \\ - 6n + 3 = 45 \\ - 6n = 45 - 3 \\ - 6 n= 42 \\ \frac{ - 6}{ - 6} n = - \frac{42}{6} \\ \boxed{n = - 7}[/tex]
Which is true of the standard form of the equation of the line graphed the value of a can be
Answer:
Consider the equation 3p - 7 + p = 13. What is the resulting equation after the first step in the solution?
Step-by-step explanation:
answer-4p - 7 = 13
help meeeeee plsssss
Piper used 1/5 meter of ribbon to create a border around a triangle if each side of the triangle is the same length how much ribbon did Piper use for each side?
Answer:
[tex]\frac{1}{15}[/tex]
Step-by-step explanation:
as said in the question that the triangle has three equal sides. i.e. this is an equilateral triangle. So,
let us consider one side of the triangle to be x . and 3x( x+x+x) should be equal to 1/5
[tex]x+x+x = \frac{1}{5}\\3x=\frac{1}{5}\\x = \frac{1}{5} / 3\\x = \frac{1}{5} * \frac{1}{3}\\x = \frac{1}{15}[/tex]
which rule best described the following pattern 5,6,15,16,25,26,35
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Answer:
Yeah im sure you wont get 3 heads for all coins it would usually be like 1 tails, 2 tails or 3 tails :/?
Step-by-step explanation:
Which equation shows the point-slope form of the line that passes through (3, 2) and has a slope of 1/3?
Answer:
Option 2: [tex]y - 2 = \frac{1}{3}(x-3)[/tex] is the correct answer.
Step-by-step explanation:
The equation of a line with given point(x1,y1) and slope m is given by:
[tex]y-y_1 = m(x-x_1)[/tex]
The line passes through (x1,y1)
Given
[tex](x_1,y_1) = (3,2)\\m = \frac{1}{3}[/tex]
=> x1 = 3
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[tex]y - 2 = \frac{1}{3}(x-3)[/tex]
Hence,
Option 2: [tex]y - 2 = \frac{1}{3}(x-3)[/tex] is the correct answer.
38,503 rounded to the nearest ten thousand
Answer:
39,000
Step-by-step explanation:
Answer:
40,000
Step-by-step explanation:
Please help fast!! Is this a function?
Answer:
It's not a function..
Step-by-step explanation:
Just take a look at it.
what did u studied about functions before doing this type of problem in math?
Does it look like it? no.
Answer:
That looks very hard .. sorry :(
Step-by-step explanation: