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bija089 [108]
2 years ago
7

Consider a circular dart board with radius one. I throw a dart at the board and record the (X,Y) position (note that the dart wi

ll land in the board, and (X,Y) are uniformly distributed on the board). 1. Find the marginal distribution of X
Mathematics
1 answer:
dybincka [34]2 years ago
7 0

Answer:

f_X (x) = \frac{2\sqrt[]{1-x^2}}{\pi}

Step-by-step explanation:

Let R be the region the dart board. Hence, the region is described by the following inequality[/tex] R=\{(x,y): x^2+y^2 \leq 1\}[/tex]. We want to find the joint density function for (X,Y). We are given that this points are uniformly distributed, hence, the joint function is related to the area of the region. Hence,

f_{X,Y)(x,y) = \frac{1}{\pi}. Note that this function integrates up to one on the region R.

\int_{R} f_{X,Y) dA = \frac{1}{\pi}\int_R 1 dA = \frac{\text{ area of R}{\pi} = \frac{\pi}{\pi}. (The area of R is pi given that it is a circle of radius 1.

Recall that the marginal distribution of X has the following pdf

f_X (x) = \int_{y} f_{X,Y}(x,y) dy. That is, fixing one value of x an integrating over the whole range of the variable Y.

Let w= \sqrt[]{1-x^2}. Then, this w represents the value of y on the boundaries of the region R. Then,

f_X (x) = \int_{-w}^{w} \frac{1}{\pi} dy = \frac{2w}{\pi} = \frac{2\sqrt[]{1-x^2}}{\pi}. This is because, when we fix a value of X, we are integrating over a vertical line in the board, so Y goes from the bottom boundary of the circle and the upper boundary of the circle, whose points are described by our variable w.

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36. The price of a new home. According to the U.S. Census
zmey [24]

Answer:

210

Step-by-step explanation:

because you use 13.% divided by the price

4 0
2 years ago
Allie sold oatmeal cookies at a stand for two days. She made 35.20 on the first day. She used 1/4 of this money to buy more supp
Tju [1.3M]
Multiply the money earned on the first day ($35.20) by 1/4.

= 35.20 * 1/4
multiply numerators

= (35.20 * 1)/4

=35.20/4
divide

= $8.80


ANSWER: $8.80 was used to buy more supplies

Hope this helps!  :)
4 0
3 years ago
A garden is designed in the shape of a rhombus formed from 4 identical 30°-60°-90° triangles. The shorter distance across the mi
Alchen [17]
        It also means that a rhombus is formed from 2 identical equilateral triangles ( 2 identical 30 - 60 - 90 triangles forms one equilateral ) and a side of that triangle measures : 30 feet. 
      Answer:
      The perimeter of the garden :  4 * 30 = 120 feet
4 0
3 years ago
Read 2 more answers
-6-5[-4-(6+12)]+(-5)
Luden [163]

Answer:

99

Step-by-step explanation:

-6-5[-4-(6+12)]+(-5)

Following PEMDAS

We work the parentheses from the inside out

-6-5[-4-(18)]+(-5)

-6-5[-22]+(-5)

Now we multiply

-6+110 -5

Now we add and subtract from left to right

104 -5

99

6 0
3 years ago
14. The area of a rectangle is 375 square inches and it’s perimeter is 80 inches. Find both the length and width.
Pavel [41]

Answer:

Length  =25 inches

Width = 15 inches

Step-by-step explanation:

Let l = length of rectangle, w = width of rectangle

Given the area and the perimeter of the rectangle, we can write:

lw = 375

2l+2w=80

So:

l+w = 80\div2

=40

l=40-w

Now, we can use substitution to find the value of w:

lw = 375

(40-w)w = 375

40w-w^2 = 375

375+w^2-40w = 0 (Quadratic equation)

(w-15)(w-25) = 0

∴ w = 15, 25

We can use substitution again to find the value of l

lw = 375

l=375\div w

=25,15

∵ Length usually refers to the longer side of a rectangle, ∴ length =25 inches and width =15 inches.

Hope this helps :)

8 0
2 years ago
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