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3 years ago

What is an algebra?

1 answer:

8 0

In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields.

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Suppose that Y has density function

zvonat [6]

I'm assuming

$f(y)=\begin{cases}ky(1-y)&\text{for }0\le y\le1\\0&\text{otherwise}\end{cases}$

(a) f(x) is a valid probability density function if its integral over the support is 1:

$\displaystyle\int_{-\infty}^\infty f(x)\,\mathrm dx=k\int_0^1 y(1-y)\,\mathrm dy=k\int_0^1(y-y^2)\,\mathrm dy=1$

Compute the integral:

$\displaystyle\int_0^1(y-y^2)\,\mathrm dy=\left(\frac{y^2}2-\frac{y^3}3\right)\bigg|_0^1=\frac12-\frac13=\frac16$

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

$\displaystyle F(y)=\int_{-\infty}^y f(t)\,\mathrm dt=\int_0^y6t(1-t)\,\mathrm dt=\begin{cases}0&\text{for }y$

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

$\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=0.95$

We have

$\displaystyle\int_{-\infty}^\varphi f(y)\,\mathrm dy=\int_0^\varphi 6y(1-y)\,\mathrm dy=(3y^2-2y^3)\bigg|_0^\varphi = 0.95$

which reduces to the cubic

3φ² - 2φ³ = 0.95

Use a calculator to solve this and find that φ ≈ 0.865.

8 0

2 years ago

15 POINTS!! HELP NOW!!

Basile [38]

The area of the bedroom is 30cm^2. Since 5m is 10 cm, 15m is 30cm.
Your answer is 15 square meters

4 0

2 years ago

Read 2 more answers

I WILL GIVE FAN AND MEDAL

Rina8888 [55]

Saving money is essential to living life. Everything costs money. Even this wifi? Not free. The device you asked this on? Not free. Saving money is important but why won't people do it? They think its hard, but it's actually easy. Step one, save one cent today, two cents tomorrow, three cents the next day...etc. That leaves you with 365 in the bank if I did the calculations right. Another way you can save is by setting a budget. Pretty easy, but you have to control what you buy and understand that buying this or that could prevent you from buying something else. The last one? Get a job. Preferably a one that gets you a higher pay without actually needing a degree, like a warehouse worker or a waiter.
These methods help you in the best way possible to save money. Good luck!

8 0

2 years ago

I cannot figure this out. Anyone? 5(-3x - 2) - (x - 3) = -4(4x + 5) + 13

ryzh [129]

Is there any choices for it

5 0

2 years ago

Describe the pattern in the following sequence and list the next three terms:

balu736 [363]

Answer:

4×2=8

8×2=16

16×2=32

32×2=64

64×2=128

128×2=256

I HOPE THIS HELPS U

5 0

2 years ago

Read 2 more answers

What is an algebra?​

What is an algebra?