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2 years ago

Evaluate the expression, (g◦f)(3), given the following functions. f(x)=x+2 and g(x)=x

Mathematics

1 answer:

azamat2 years ago

5 0

$f(x) = x + 2\\g(x) = x\\(g \circ f) = g(f(x)) = x + 2\\g \circ f)(3) = 3 + 2 = 5$

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Let X ~ N(0, 1) and Y = eX. Y is called a log-normal random variable.

Cloud [144]

If $F_Y(y)$ is the cumulative distribution function for $Y$ , then

$F_Y(y)=P(Y\le y)=P(e^X\le y)=P(X\le\ln y)=F_X(\ln y)$

Then the probability density function for $Y$ is $f_Y(y)={F_Y}'(y)$ :

$f_Y(y)=\dfrac{\mathrm d}{\mathrm dy}F_X(\ln y)=\dfrac1yf_X(\ln y)=\begin{cases}\frac1{y\sqrt{2\pi}}e^{-\frac12(\ln y)^2}&\text{for }y>0\\0&\text{otherwise}\end{cases}$

The $n$ th moment of $Y$ is

$E[Y^n]=\displaystyle\int_{-\infty}^\infty y^nf_Y(y)\,\mathrm dy=\frac1{\sqrt{2\pi}}\int_0^\infty y^{n-1}e^{-\frac12(\ln y)^2}\,\mathrm dy$

Let $u=\ln y$ , so that $\mathrm du=\frac{\mathrm dy}y$ and $y^n=e^{nu}$ :

$E[Y^n]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{nu}e^{-\frac12u^2}\,\mathrm du=\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{nu-\frac12u^2}\,\mathrm du$

Complete the square in the exponent:

$nu-\dfrac12u^2=-\dfrac12(u^2-2nu+n^2-n^2)=\dfrac12n^2-\dfrac12(u-n)^2$

$E[Y^n]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{\frac12(n^2-(u-n)^2)}\,\mathrm du=\frac{e^{\frac12n^2}}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac12(u-n)^2}\,\mathrm du$

But $\frac1{\sqrt{2\pi}}e^{-\frac12(u-n)^2}$ is exactly the PDF of a normal distribution with mean $n$ and variance 1; in other words, the 0th moment of a random variable $U\sim N(n,1)$ :

$E[U^0]=\displaystyle\frac1{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-\frac12(u-n)^2}\,\mathrm du=1$

so we end up with

$E[Y^n]=e^{\frac12n^2}$

3 0

2 years ago

Given a scalene triangle with whole number angle measurements, what is the greatest sum possible of the two smallest angles?

Olegator [25]

SOLUTION

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Define scalene triangle

A Scalene Triangle is any triangle with unequal sides. This means that, in a scalene triangle, all of the three sides and angles are different lengths, just like in the illustration below. This also means that each angle has to be different.

STEP 2: Get the greatest sum possible of the two smallest angles

Since the measures of the angles of the triangle are different whole numbers , for the sum of the two angles to be the least possible, one of the angles must be smallest whole number i.e 1°

Now, the next smallest angle will be the next angle of a whole number that is not 1 degree, i.e, 2 degrees.

Thus, the greatest possible sum of the measures of two smallest angles will be:

$1^{\circ}+2^{\circ}=3^{\circ}$

Hence, the greatest sum possible of the two smallest angles is 3 degrees

7 0

1 year ago

Will do a brainly! 4/7=18/31.5

german

Answer:

True

Step-by-step explanation:

4 ÷ 7 = 0.5714
18 ÷ 31.5 = 0.5714
Because they are the same result, the two fractions are equal. Therefore, the equation is true.

I hope this helps!

3 0

3 years ago

3/unknow ÷5=1/10 what in the unknown?

saul85 [17]

3/x÷5=1/10
3÷5=1/10*x
6/10=1/10*x
6=x

5 0

2 years ago

Read 2 more answers

Select the correct answer.

Reika [66]

Answer:

A and C

Step-by-step explanation:

8 0

3 years ago

Evaluate the​ expression, ​(g◦​f)(3​), given the following functions. ​f(x)=x+2 and ​g(x)=x

Evaluate the expression, (g◦f)(3), given the following functions. f(x)=x+2 and g(x)=x