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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- x - 2 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- x - 2 y}\, dx\, dy = 1\)

Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} c e^{- 2 y} - c e^{- 3 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{2} e^{- 2 y} + \frac{c}{3} e^{- 3 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{6}\right) + \lim_{b \to \infty}\left(- \frac{c}{2} e^{- 2 b} + \frac{c}{3} e^{- 3 b}\right)\)

\(\displaystyle \frac{c}{6} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 6\)

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

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

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

Now let's set up the double integral.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- x - 19 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- x - 19 y}\, dx\, dy = 1\)

Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} c e^{- 19 y} - c e^{- 20 y}\, dy = 1\)
To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b\to \infty}\left(- \frac{c}{19} e^{- 19 y} + \frac{c}{20} e^{- 20 y}\right) \Big|_{ 0 }^{ 19 }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{380}\right) + \lim_{b\to \infty}\left(- \frac{c}{19 e^{361}} + \frac{c}{20 e^{380}}\right)\)

\(\displaystyle \frac{c}{380} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 380\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 7 x - 14 y}\, dx\, dy\)
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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 7 x - 14 y}\, dx\, dy = 1\)


Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{7} e^{- 14 y} - \frac{c}{7} e^{- 21 y}\, dy = 1\)

To evaluate this, we need to use the limit definition
\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{98} e^{- 14 y} + \frac{c}{147} e^{- 21 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{294}\right) + \lim_{b \to \infty}\left(- \frac{c}{98} e^{- 14 b} + \frac{c}{147} e^{- 21 b}\right)\)

\(\displaystyle \frac{c}{294} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 294\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 5 x - 6 y}\, dx\, dy\)
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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 5 x - 6 y}\, dx\, dy = 1\)


Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{5} e^{- 6 y} - \frac{c}{5} e^{- 11 y}\, dy = 1\)
To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{30} e^{- 6 y} + \frac{c}{55} e^{- 11 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{66}\right) + \lim_{b \to \infty}\left(- \frac{c}{30} e^{- 6 b} + \frac{c}{55} e^{- 11 b}\right)\)

\(\displaystyle \frac{c}{66} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 66\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 12 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 9 x - 12 y}\, dx\, dy = 1\)


Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{9} e^{- 12 y} - \frac{c}{9} e^{- 21 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{108} e^{- 12 y} + \frac{c}{189} e^{- 21 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{252}\right) + \lim_{b \to \infty}\left(- \frac{c}{108} e^{- 12 b} + \frac{c}{189} e^{- 21 b}\right)\)

\(\displaystyle \frac{c}{252} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 252\)

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

Let's find what the bounds are for both \(\displaystyle \uptext{x}\), and \(\displaystyle \uptext{y}\)
We look at the p.d.f to see that the bounds for \(\displaystyle \uptext{x}\) are, \(\displaystyle 0< x< y\), and for \(\displaystyle \uptext{y}\), \(\displaystyle 0< y< \infty\)

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 6 x - 10 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 6 x - 10 y}\, dx\, dy = 1\)


Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{6} e^{- 10 y} - \frac{c}{6} e^{- 16 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{60} e^{- 10 y} + \frac{c}{96} e^{- 16 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{160}\right) + \lim_{b \to \infty}\left(- \frac{c}{60} e^{- 10 b} + \frac{c}{96} e^{- 16 b}\right)\)

\(\displaystyle \frac{c}{160} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 160\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 13 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 10 x - 13 y}\, dx\, dy = 1\)


Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{10} e^{- 13 y} - \frac{c}{10} e^{- 23 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{130} e^{- 13 y} + \frac{c}{230} e^{- 23 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{299}\right) + \lim_{b \to \infty}\left(- \frac{c}{130} e^{- 13 b} + \frac{c}{230} e^{- 23 b}\right)\)

\(\displaystyle \frac{c}{299} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 299\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 8 x - 9 y}\, dx\, dy\)
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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 8 x - 9 y}\, dx\, dy = 1\)

Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} \frac{c}{8} e^{- 9 y} - \frac{c}{8} e^{- 17 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{72} e^{- 9 y} + \frac{c}{136} e^{- 17 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{153}\right) + \lim_{b \to \infty}\left(- \frac{c}{72} e^{- 9 b} + \frac{c}{136} e^{- 17 b}\right)\)

\(\displaystyle \frac{c}{153} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 153\)

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

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

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

Now let's set up the double integral

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

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.

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

Now evaluate the double integral

\(\displaystyle \int_{0}^{\infty} c e^{- y} - c e^{- 2 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- c e^{- y} + \frac{c}{2} e^{- 2 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{2}\right) + \lim_{b \to \infty}\left(- c e^{- b} + \frac{c}{2} e^{- 2 b}\right)\)

\(\displaystyle \frac{c}{2} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 2\)

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

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

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

Now let's set up the double integral

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 2 x - 10 y}\, dx\, dy\)

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.

\(\displaystyle \int_{0}^{\infty}\int_{0}^{y} c e^{- 2 x - 10 y}\, dx\, dy = 1\)


Now evaluate the double integral


\(\displaystyle \int_{0}^{\infty} \frac{c}{2} e^{- 10 y} - \frac{c}{2} e^{- 12 y}\, dy = 1\)

To evaluate this, we need to use the limit definition

\(\displaystyle \lim_{b \to \infty}\left(- \frac{c}{20} e^{- 10 y} + \frac{c}{24} e^{- 12 y}\right) \Big|_{ 0 }^{ b }\)

\(\displaystyle - \lim_{b \to \infty}\left(- \frac{c}{120}\right) + \lim_{b \to \infty}\left(- \frac{c}{20} e^{- 10 b} + \frac{c}{24} e^{- 12 b}\right)\)

\(\displaystyle \frac{c}{120} = 1\)

Now we simply solve for \(\displaystyle \uptext{c}\)

\(\displaystyle c = 120\)