Probability Theory : Conditional Distributions and Independence

Study concepts, example questions & explanations for Probability Theory

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Example Questions

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Example Question #1 : Conditional Distributions And Independence

Let , and  be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.

Determine the value of

Possible Answers:

Correct answer:

Explanation:

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 #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

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 #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 .

Possible Answers:

Correct answer:

Explanation:

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 #2 : Conditional Distributions And Independence

Let , and  be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.


Determine the value of

Possible Answers:

Correct answer:

Explanation:

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 : Conditional Distributions And Independence

Let , and  be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.


Determine the value of .

Possible Answers:

Correct answer:

Explanation:

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 #4 : Conditional Distributions And Independence

Let , and  be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.


Determine the value of

Possible Answers:

Correct answer:

Explanation:

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 #8 : 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 .

Possible Answers:

Correct answer:

Explanation:

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 #5 : Conditional Distributions And Independence

Let , and  be the lifespans (in hours) of two electronic devices, and their joint probability mass function is given below.


Determine the value of .

Possible Answers:

Correct answer:

Explanation:

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 #10 : 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 .

Possible Answers:

Correct answer:

Explanation:

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

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