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Svetlanka [38]
3 years ago
11

(Uniform) Suppose X follows a continuous uniform distribution from 3 to 9. (a) Write down the PDF of X. (b) Find P(X ≤ 6). Round

your answer to 4 decimals. (c) Find P(4 < X ≤ 6). Round your answer to 4 decimals. (d) Determine the conditional probability P(X > 4 | X ≤ 6). Round your answer to 4 decimals
Mathematics
1 answer:
defon3 years ago
7 0

Answer:

a)  P(x) = {0.1667 for x in (3,9) ,   0 otherwise}

b) P(x≤6) = 0.5

c) P(4 < x ≤ 6) = 0.3333

d) P(X > 4 | X ≤ 6) = 0.3333 or 1 <em>*see below*</em>

<em />

Step-by-step explanation:

a)

Since it is a uniform distribution you can imagine it as a rectangle which extends from a = 3 to b = 9 and whose total area is 1.

Therefore the height must be equal to the inverse of the side:

P(X) = 1/(b-a) = 1/(9-3)= 1/6       for every x in (3,9)  and 0 otherwise

P(X) = 0.1667

b)

To find this probability we must find the area of the rectangle which extends from 3 to 6 and whose height is equal to P(x)

That is:

P(X ≤ 6) = (6-3)/(9-3) = 3/6 = 1/2 = 0.5

That make sense since 6 is half way between 3 and 9

c)

Similar to b) but now we consider tha base of the rectangle from 4 to 6:

P(4 < X ≤ 6) = (6-4)/(9-3) = 2/6 = 1/3 = 0.3333

d)

If  '' | '' means " or ", the probability is one, since every number in the range (3,9) is either higher than 4 or less than 6.

If " | " means " and " the answer is the same as in c , that is because 4<x≤6 is the set of numbers which are higher than 4 and at most 6 which can be expressed as x>4 and x≤6

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Let A = {x : 2x² + 3x - 2 = 0} and B = {x : x² + 3x - 4 = 0 } find (A U B) × (A Π B)​
ivann1987 [24]

Given :

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  • B = {x : x² + 3x - 4 = 0 }

To find :

  • (A U B ) × (A Π B )

Solution :-

<u>The </u><u>first </u><u>set </u><u>is </u><u>,</u>

  • A ={x : 2x² + 3x - 2 = 0}

<u>Solving</u><u> </u><u>the </u><u>Quadratic</u><u> equation</u><u> </u><u>,</u>

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  • (2x -1) ( x + 2) = 0
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<u>Hence</u><u> </u><u>,</u>

  • A ={0.5, -2}

<u>The </u><u>second</u><u> </u><u>set </u><u>is </u><u>,</u>

  • B ={ x :x² + 3x - 4 = 0 }

<u>Solving</u><u> the</u><u> Quadratic</u><u> equation</u><u> </u><u>,</u>

  • x² + 3x - 4 = 0
  • x² + 4x - x - 4 = 0
  • x( x + 4)-1 ( x +4) = 0
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<u>Hence</u><u> </u><u>,</u>

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<u>Now </u><u>,</u>

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Since AΠ B is a null set , hence ,

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To show that the identity is true, we need to show that the left hand side is equal to right hand side or vice versa.

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According to Pythagoras theorem, we know that x²+y² = r² in a right angled triangle. Coverting this to polar form, we have:

x = rcostheta

y = rsintheta

Substituting into the Pythagoras firnuka we have

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r²(cos²theta+sin²theta) = r²

(cos²theta+sin²theta) = 1

sin²theta = 1 - cos²theta

sin²x = 1-cos²x ... 2

Substituting equation 2 into 1 we have;

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