Multiple Random Variables - Probability Theory

Card 0 of 48

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

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

To evaluate this, we need to use the limit definition

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

$$\frac{c}{6}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

In order to find the value of $\up$\text{c}$$ , we need to take find the double integral of the function.

Let's find what the bounds are for both $\up$\text{x}$$ , and $\up$\text{y}$$ .

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

Now let's set up the double integral.

$\int_${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.

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

Now evaluate the double integral

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

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

$$\frac{c}{380}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

In order to find the value of $\up$\text{c}$$ , we need to take find the double integral of the function.

Let's find what the bounds are for both $\up$\text{x}$$ , and $\up$\text{y}$$ .

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{294}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{66}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{252}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{160}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{299}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{153}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

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

To evaluate this, we need to use the limit definition

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

$$\frac{c}{2}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{120}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

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

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$

Now evaluate the double integral

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

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

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

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

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$

Now evaluate the double integral

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

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

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

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

To evaluate this, we need to use the limit definition

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

$$\frac{c}{6}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

In order to find the value of $\up$\text{c}$$ , we need to take find the double integral of the function.

Let's find what the bounds are for both $\up$\text{x}$$ , and $\up$\text{y}$$ .

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

Now let's set up the double integral.

$\int_${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.

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

Now evaluate the double integral

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

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

$$\frac{c}{380}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

In order to find the value of $\up$\text{c}$$ , we need to take find the double integral of the function.

Let's find what the bounds are for both $\up$\text{x}$$ , and $\up$\text{y}$$ .

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{294}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{66}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{252}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{160}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{299}$ = 1$

Now we simply solve for $\up$\text{c}$$

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

Determine the value of $\up$\text{c}$$ .

Tap to see back →

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

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

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

Now let's set up the double integral

$\int_${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.

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

Now evaluate the double integral

$\int_${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

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

$$\frac{c}{153}$ = 1$

Now we simply solve for $\up$\text{c}$$