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Probability Theory : Conditional Distributions and Independence

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Example Question #11 : Conditional Distributions And Independence

Let $\uptext{X}$ $\displaystyle \uptext{X}$, and $\uptext{Y}$ $\displaystyle \uptext{Y}$ be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.

$\operatorname{f_{X,Y}}{\left (x,y \right )} = \begin{cases} c e^{- 10 x - 10 y} & 0<x<y<\infty \\ 0, & \text{Otherwise} \end{cases}$ $\displaystyle \operatorname{f_{X,Y}}{\left (x,y \right )} = \begin{cases} c e^{- 10 x - 10 y} & 0< x< y< \infty \\ 0, & \text{Otherwise} \end{cases}$

Determine the value of $\uptext{c}$ $\displaystyle \uptext{c}$

Possible Answers:

c = 400 $\displaystyle c = 400$

c = 1 $\displaystyle c = 1$

c = 50 $\displaystyle c = 50$

$c = \frac{200}{3}$ $\displaystyle c = \frac{200}{3}$

c = 200 $\displaystyle c = 200$

Correct answer:

c = 200 $\displaystyle c = 200$

Explanation:

In order to find the value of $\uptext{c}$ $\displaystyle \uptext{c}$, we need to take find the double integral of the function

Let's find what the bounds are for both $\uptext{x}$ $\displaystyle \uptext{x}$, and $\uptext{y}$ $\displaystyle \uptext{y}$

We look at the p.d.f to see that the bounds for $\uptext{x}$ $\displaystyle \uptext{x}$ are, 0<x<y $\displaystyle 0< x< y$, and for $\uptext{y}$ $\displaystyle \uptext{y}$, $0<y<\infty$ $\displaystyle 0< y< \infty$

Now let's set up the double integral

$\int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 10 y}\, dx\, dy$ $\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 10 y}\, dx\, dy$

Before we evaluate it, we need to remember to set the double integral equal to one, since we are essentially solving for the c.d.f.

$\int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 10 y}\, dx\, dy = 1$ $\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 10 y}\, dx\, dy = 1$

Now evaluate the double integral

$\int_{0}^{\infty} \frac{c}{10} e^{- 10 y} - \frac{c}{10} e^{- 20 y}\, dy = 1$ $\displaystyle \int_{0}^{\infty} \frac{c}{10} e^{- 10 y} - \frac{c}{10} e^{- 20 y}\, dy = 1$

To evaluate this, we need to use the limit definition

$\lim_{b \to \infty}\left(- \frac{c}{100} e^{- 10 y} + \frac{c}{200} e^{- 20 y}\right) \Big|_{ 0 }^{ b }$ $\displaystyle \lim_{b \to \infty}\left(- \frac{c}{100} e^{- 10 y} + \frac{c}{200} e^{- 20 y}\right) \Big|_{ 0 }^{ b }$

$- \lim_{b \to \infty}\left(- \frac{c}{200}\right) + \lim_{b \to \infty}\left(- \frac{c}{100} e^{- 10 b} + \frac{c}{200} e^{- 20 b}\right)$ $\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{200}\right) + \lim_{b \to \infty}\left(- \frac{c}{100} e^{- 10 b} + \frac{c}{200} e^{- 20 b}\right)$

$\frac{c}{200} = 1$ $\displaystyle \frac{c}{200} = 1$

Now we simply solve for $\uptext{c}$ $\displaystyle \uptext{c}$

c = 200 $\displaystyle c = 200$

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Example Question #11 : Conditional Distributions And Independence

$\operatorname{f_{X,Y}}{\left (x,y \right )} = \begin{cases} c e^{- 9 x - 11 y} & 0<x<y<\infty \\ 0, & \text{Otherwise} \end{cases}$ $\displaystyle \operatorname{f_{X,Y}}{\left (x,y \right )} = \begin{cases} c e^{- 9 x - 11 y} & 0< x< y< \infty \\ 0, & \text{Otherwise} \end{cases}$

Determine the value of $\uptext{c}$ $\displaystyle \uptext{c}$.

Possible Answers:

c = 55 $\displaystyle c = 55$

c = 220 $\displaystyle c = 220$

$c = \frac{220}{3}$ $\displaystyle c = \frac{220}{3}$

c = 1 $\displaystyle c = 1$

c = 440 $\displaystyle c = 440$

Correct answer:

c = 220 $\displaystyle c = 220$

Explanation:

In order to find the value of $\uptext{c}$ $\displaystyle \uptext{c}$, we need to take find the double integral of the function

Let's find what the bounds are for both $\uptext{x}$ $\displaystyle \uptext{x}$, and $\uptext{y}$ $\displaystyle \uptext{y}$

We look at the p.d.f to see that the bounds for $\uptext{x}$ $\displaystyle \uptext{x}$ are, 0<x<y $\displaystyle 0< x< y$, and for $\uptext{y}$ $\displaystyle \uptext{y}$, $0<y<\infty$ $\displaystyle 0< y< \infty$

Now let's set up the double integral

$\int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 11 y}\, dx\, dy$ $\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 11 y}\, dx\, dy$

Before we evaluate it, we need to remember to set the double integral equal to one, since we are essentially solving for the c.d.f.

$\int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 11 y}\, dx\, dy = 1$ $\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 11 y}\, dx\, dy = 1$

Now evaluate the double integral

$\int_{0}^{\infty} \frac{c}{9} e^{- 11 y} - \frac{c}{9} e^{- 20 y}\, dy = 1$ $\displaystyle \int_{0}^{\infty} \frac{c}{9} e^{- 11 y} - \frac{c}{9} e^{- 20 y}\, dy = 1$

To evaluate this, we need to use the limit definition

$\lim_{b \to \infty}\left(- \frac{c}{99} e^{- 11 y} + \frac{c}{180} e^{- 20 y}\right) \Big|_{ 0 }^{ b }$ $\displaystyle \lim_{b \to \infty}\left(- \frac{c}{99} e^{- 11 y} + \frac{c}{180} e^{- 20 y}\right) \Big|_{ 0 }^{ b }$

$- \lim_{b \to \infty}\left(- \frac{c}{220}\right) + \lim_{b \to \infty}\left(- \frac{c}{99} e^{- 11 b} + \frac{c}{180} e^{- 20 b}\right)$ $\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{220}\right) + \lim_{b \to \infty}\left(- \frac{c}{99} e^{- 11 b} + \frac{c}{180} e^{- 20 b}\right)$

$\frac{c}{220} = 1$ $\displaystyle \frac{c}{220} = 1$

Now we simply solve for $\uptext{c}$ $\displaystyle \uptext{c}$

c = 220 $\displaystyle c = 220$

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Probability Theory : Conditional Distributions and Independence

Study concepts, example questions & explanations for Probability Theory

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Example Question #11 : Conditional Distributions And Independence

Example Question #11 : Conditional Distributions And Independence

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