All Probability Theory Resources
Example Questions
Example Question #1 : Multiple Random Variables
Let
Determine the value of
In order to find the value of
, we need to take find the double integral of the functionLet's find what the bounds are for both
, andWe look at the p.d.f to see that the bounds for
are, , and for ,Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #2 : Multiple Random Variables
Let
, and be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.
Determine the value of
In order to find the value of
, we need to take find the double integral of the function.Let's find what the bounds are for both
, and .We look at the p.d.f to see that the bounds for
are, , and for , .Now let's set up the double integral.
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #3 : Multiple Random Variables
Let
, and be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.
Determine the value of
.
In order to find the value of
, we need to take find the double integral of the function.Let's find what the bounds are for both
We look at the p.d.f to see that the bounds for
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #4 : Multiple Random Variables
Let
, and be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.
Determine the value of
.
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #5 : Multiple Random Variables
Let
, and be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.Determine the value of
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #6 : Multiple Random Variables
Let
Determine the value of .
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #7 : Multiple Random Variables
Let
Determine the value of
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #8 : Multiple Random Variables
Let
Determine the value of .
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #9 : Multiple Random Variables
Let
Determine the value of .
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
Example Question #10 : Multiple Random Variables
Let
Determine the value of
.
In order to find the value of
Let's find what the bounds are for both , and
We look at the p.d.f to see that the bounds for are, , and for ,
Now let's set up the double integral
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.
Now evaluate the double integral
To evaluate this, we need to use the limit definition
Now we simply solve for
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