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ikadub [295]
3 years ago
10

A filtration process removes a random proportion of particulates in water to which it is applied. Suppose that a sample of water

is subjected to this process twice. Let x1 be the proportion of the particulates that are removed by the first pass. Let X2 be the proportion of what remains after the first pass that is removed by the second pass. Assume that X1 and X2 are independent random variables with common pdf. f(x) = 4x3, for 0 < x <1 and f(x) = 0 otherwise. Let Y be the proportion of the original particulates that remain in the sample after two passes. Then Y = (1 - X1)(1 - X2). Find E(Y).
Mathematics
1 answer:
Doss [256]3 years ago
3 0

Answer:

E(Y)=\frac{1}{25}

Step-by-step explanation:

Let's start defining the random variables for this exercise :

X_{1}: '' The proportion of the particulates that are removed by the first pass ''

X_{2}: '' The proportion of what remains after the first pass that is removed by the second pass ''

Y: '' The proportion of the original particulates that remain in the sample after two passes ''

We know the relation between the random variables :

Y=(1-X_{1})(1-X_{2})

We also assume that X_{1} and X_{2} are independent random variables with common pdf.

The probability density function for both variables is f(x)=4x^{3} for 0 and f(x)=0 otherwise.

The first step to solve this exercise is to find the expected value for X_{1} and X_{2}.

Because the variables have the same pdf we write :

E(X_{1})= E(X_{2})=E(X)

Using the pdf to calculate the expected value we write :

E(X)=\int\limits^a_b {xf(x)} \, dx

Where a= ∞ and b= - ∞ (because we integrate in the whole range of the random variable). In this case, we will integrate between 0 and 1 ⇒

Using the pdf we calculate the expected value :

E(X)=\int\limits^1_0 {x4x^{3}} \, dx=\int\limits^1_0 {4x^{4}} \, dx=\frac{4}{5}

⇒ E(X)=E(X_{1})=E(X_{2})=\frac{4}{5}

Now we need to use some expected value properties in the expression of Y ⇒

Y=(1-X_{1})(1-X_{2}) ⇒

Y=1-X_{2}-X_{1}+X_{1}X_{2}

Applying the expected value properties (linearity and expected value of a constant) ⇒

E(Y)=E(1)-E(X_{2})-E(X_{1})+E(X_{1}X_{2})

Using that X_{1} and X_{2} have the same expected value E(X) and given that X_{1} and X_{2} are independent random variables we can write E(X_{1}X_{2})=E(X_{1})E(X_{2})   ⇒

E(Y)=E(1)-E(X)-E(X)+E(X_{1})E(X_{2}) ⇒

E(Y)=E(1)-2E(X)+[E(X)]^{2}

Using the value of E(X) calculated :

E(Y)=1-2(\frac{4}{5})+(\frac{4}{5})^{2}=\frac{1}{25}

E(Y)=\frac{1}{25}

We find that the expected value of the variable Y is E(Y)=\frac{1}{25}

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A family has two children. If the genders of these children are listed in the order they are born, there are four possible outco
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Answer:

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Step-by-step explanation:

The possible outcomes of this event are: BB, BG, GB and GG.

There is one out of four events (GG) in which there are two girls (X=2).

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2 years ago
A circle in the xy-plane has a diameter with endpoints whose coordinates are negative 1 comma negative 3 and 7 comma 3. if the p
DerKrebs [107]

<span>First we look for the coordinates of the center of the circle. For this, we use the following formula:</span>

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<span> (x1, y1) = (- 1, -3)</span>

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<span> Substituting values we have:</span>

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<span> ((6) / 2, (0) / 2)</span>

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<span> We are now looking for the diameter of the circle. For this we use the formula of distance between points:</span>

<span> d = root ((x2-x1) ^ 2 + (y2-y1) ^ 2)</span>

<span> Substituting values:</span>

<span> d = root ((7 - (- 1)) ^ 2 + (3 - (- 3)) ^ 2)</span>

<span> d = 10</span>

<span> Then, the radius of the circle is:</span>

<span> r = d / 2 = 10/2</span>

<span> r = 5</span>

<span> The circle equation will be</span>

<span> (x-h) ^ 2 + (y-k) ^ 2 = r ^ 2</span>

<span> Where,</span>

<span> (h, k) = (3, 0)</span>

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<span> (x-3) ^ 2 + (y-0) ^ 2 = 5 ^ 2</span>

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<span><span> b = root (16)</span></span>

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<span><span> Answer</span></span>

<span><span> the value of b is</span></span>

<span><span> <span>b = 4</span></span></span>


8 0
3 years ago
Read 2 more answers
Can someone please help me?
garik1379 [7]
The answer is b). (0,0)
8 0
3 years ago
Renaldo will write 3/20 as a decimal. Which of the following methods should he use?
makvit [3.9K]

ANSWER

1. Multiply the fraction by 5/5 to get a denominator of 100 and then write the numerator as hundredths using a decimal point.

EXPLANATION

Renaldo wants to write

\frac{3}{20}

as a decimal.

He needs a denominator of 100, so he can multiply by

\frac{5}{5}

to obtain:

\frac{3}{20}  =  \frac{3 \times 5}{20 \times 5}

This will be

=  \frac{15}{100}

He can now obtain the decimal equivalent as

= 0.15

The correct choice is option 1.

3 0
3 years ago
Read 2 more answers
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