ผลต่างระหว่างรุ่นของ "Probstat/week8 practice 1"

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* 3.3 Let <math>F(x)</math> be <math>X</math>'s cumulative distribution function.  Find <math>F(x)</math>.
 
* 3.3 Let <math>F(x)</math> be <math>X</math>'s cumulative distribution function.  Find <math>F(x)</math>.
 
* 3.4 Plot <math>F(x)</math>
 
* 3.4 Plot <math>F(x)</math>
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4. (FCP. Example 1a.) Suppose that ''X'' is a continuous random variable whose pdf is given by
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<center><math>f(x) = \left\{\begin{array}{ll}C(9x - 3x^2) \ \ \  & 0 < x < 3 \\ 0 & \mbox{otherwise.}\end{array}\right.</math></center>
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* 4.1 Find <math>C</math>.
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* 4.2 Find <math>P\{ X > 1 \}</math>.
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5. (FCP. Example 1b.)  The time (in hours) that a computer functions before breaking down is a continuous random variable <math>X</math> with probability density function given by
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<center><math>f(x) = \left\{\begin{array}{ll}\lambda e^{-x/100} \ \ \  & x\geq 0\\ 0 & x < 0\end{array}\right.</math></center>
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* 5.1 Find the value of <math>\lambda</math>
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* 5.2 What is the probability that a computer will function between 50 and 100 hours before breaking down?
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* 5.3 What is the probability that it will function less than 100 hours?
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'''Hint:''' Note that
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<center>
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<math>\int e^{-x} dx = -e^{-x} + C</math>
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</center>
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6. A random variable <math>X</math> has pdf defined by
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<center><math>f(x) = \left\{\begin{array}{ll}C/x^2 \ \ \  & x > 20 \\ 0 & \mbox{otherwise.}\end{array}\right.</math></center>
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* 6.1 Find <math>C</math>.
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* 6.2 Find its cdf <math>F(x)</math> and plot it.
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* 6.3 What is <math>P\{ X > 30 \}</math>?
  
 
== Expectation ==
 
== Expectation ==
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<math>\mathrm{E}[X] = \int_{-\infty}^{\infty} xf(x)dx</math>
 
<math>\mathrm{E}[X] = \int_{-\infty}^{\infty} xf(x)dx</math>
 
</center>
 
</center>
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1. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 1 in the previous section.
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2. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 2 in the previous section.
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3. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 3 in the previous section.
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4. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 4 in the previous section.
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5. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 5 in the previous section.
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'''Hint:''' Note that
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<center>
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<math>\int x e^{-x} dx = -e^{-x}(x+1) + C</math>
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</center>
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6. Find <math>\mathrm{E}[X]</math> for continuous random variable <math>X</math> defined in question 6 in the previous section.
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7. Find <math>\mathrm{E}[X^2]</math> and Var(<math>X</math>) for continuous random variable <math>X</math> defined in question 1 in the previous section.
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8. Find <math>\mathrm{E}[X^2]</math> and Var(<math>X</math>) for continuous random variable <math>X</math> defined in question 2 in the previous section.
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9. Find <math>\mathrm{E}[X^2]</math> and Var(<math>X</math>) for continuous random variable <math>X</math> defined in question 3 in the previous section.

รุ่นแก้ไขปัจจุบันเมื่อ 03:26, 7 ตุลาคม 2557

This is part of probstat

This exercise considers continuous random variables.

Probability

1. Let random variable is a continuous random variable which is a real number chosen uniformly from the range (10,50). Let denote its probability distribution function.

  • 1.1 Describe .
  • 1.2 What is .
  • 1.3 What is . (Note that this is a conditional probability.)

2. Let random variable is a continuous random variable which is a real number chosen uniformly from the range (a,b). Let denote its probability distribution function.

  • 2.1 Describe .
  • 2.2 Find the value of such that .

Definition: A cumulative distribution function (or cdf in short) for a continuous random variable whose probability distribution function is is defined to be

3. Let random variable is a continuous random variable whose probability distribution function is given by

  • 3.1 Find the value of c.
  • 3.2 Plot pdf of X.
  • 3.3 Let be 's cumulative distribution function. Find .
  • 3.4 Plot

4. (FCP. Example 1a.) Suppose that X is a continuous random variable whose pdf is given by

  • 4.1 Find .
  • 4.2 Find .

5. (FCP. Example 1b.) The time (in hours) that a computer functions before breaking down is a continuous random variable with probability density function given by

  • 5.1 Find the value of
  • 5.2 What is the probability that a computer will function between 50 and 100 hours before breaking down?
  • 5.3 What is the probability that it will function less than 100 hours?

Hint: Note that

6. A random variable has pdf defined by

  • 6.1 Find .
  • 6.2 Find its cdf and plot it.
  • 6.3 What is ?

Expectation

Definition: If is a continuous random variable whose probability distribution function (pdf) is , we define the expected value of to be

1. Find for continuous random variable defined in question 1 in the previous section.

2. Find for continuous random variable defined in question 2 in the previous section.

3. Find for continuous random variable defined in question 3 in the previous section.

4. Find for continuous random variable defined in question 4 in the previous section.

5. Find for continuous random variable defined in question 5 in the previous section.

Hint: Note that

6. Find for continuous random variable defined in question 6 in the previous section.

7. Find and Var() for continuous random variable defined in question 1 in the previous section.

8. Find and Var() for continuous random variable defined in question 2 in the previous section.

9. Find and Var() for continuous random variable defined in question 3 in the previous section.