Show f is a probability density function

It's essentially the same as using the derivative to show that f(x) is always increasing or decreasing. Good job!In summary, to show that f(x) is a density, we need to prove that f(x) is always greater than or equal to 0 and that the integral of f(x) from -infinity to infinity is equal to 1. We can prove the latter by taking the integral of f(x), which is equal to 1. To prove the former, we can show that the derivative of f(x) is always greater than or equal to 0, which we can do by considering the boundaries of the interval -1<=x<=1 and using the fact that u is between -1 and 1.
  • #1
saizen21
3
0

Homework Statement



Let f(x) = (1 + ux)/2 for -1<= x <= 1 and 0 otherwise . where -1<= u <= 1

a) show f is a density


Homework Equations



TO show
1. f(x) >= 0

2. intergeral f (from -infinity to infinity) = 1

The Attempt at a Solution



I have done 2. and proved that it is 1 by taking the intergeral of f.

However, how do u show f >= 0.

i have found the dervavtive of f to be u / 2 for -1<= x <= 1 and 0 otherwise.

I dun know where to go?
 
Physics news on Phys.org
  • #2
If you've shown f'(x)=u/2 then that means the max and min of f(x) must be at x=(-1) or x=1, the boundaries of your interval. Can you take it from there?
 
  • #3
is this also possible?

1<=x<=1
-1<=ux<=1
0<=1+ux<=2
0<=(1+ux)/2<=1
therefore its always >= 0
 
  • #4
saizen21 said:
is this also possible?

1<=x<=1
-1<=ux<=1
0<=1+ux<=2
0<=(1+ux)/2<=1
therefore its always >= 0

That works too.
 

Related to Show f is a probability density function

1. What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the relative likelihood of a continuous random variable taking on a specific value. It is used to model the probability distribution of a continuous random variable.

2. How is a PDF different from a probability mass function?

A PDF is used to describe the probability distribution of a continuous random variable, while a probability mass function (PMF) is used for discrete random variables. In other words, a PDF is used to model the probabilities of an infinite number of outcomes, while a PMF is used for a finite number of outcomes.

3. What are the properties of a valid PDF?

A valid PDF must have the following properties:

  • Non-negative values: The PDF must always have non-negative values, as probabilities cannot be negative.
  • Total area under the curve equals 1: The total area under the curve of a PDF must equal 1, as the sum of all possible outcomes must equal 1.
  • Probability of a single outcome is 0: The probability of a single outcome occurring is 0, as a continuous random variable can take on an infinite number of values.

4. How is a PDF used in statistics?

A PDF is used in statistics to calculate the probability of a continuous random variable falling within a specific range of values. This allows for the calculation of probabilities for events that have a range of possible outcomes, rather than just a single outcome.

5. Can any function be a PDF?

No, not every function can be a PDF. A valid PDF must meet certain criteria, such as being non-negative and having a total area under the curve equal to 1. Additionally, a PDF must be able to describe the relative likelihood of all possible outcomes within a given range, which may not be possible for certain functions.

Similar threads

Probability density function of Z=X+Y and Z=XY
    • Calculus and Beyond Homework Help
Replies
19
Views
1K
Solve this problem that involves a probability density function
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
How should I use the Jacobi equation to show this?
    • Calculus and Beyond Homework Help
Replies
4
Views
761
Finding marginal distribution of 2d of probability density function
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Probability Density of ##x## (Wiener Process)
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Solve the problem involving probability density function
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
Show uniqueness in Dirichlet problem in unit disk
    • Calculus and Beyond Homework Help
Replies
6
Views
472
Is It Possible to Invert a Homotopy?
    • Calculus and Beyond Homework Help
Replies
5
Views
401
For this Partial Derivative -- Why are different results obtained?
    • Calculus and Beyond Homework Help
Replies
23
Views
1K
How should I use the Jacobi equation to determine the nature of this?
    • Calculus and Beyond Homework Help
Replies
5
Views
691

Hot Threads

Recent Insights

Back
Top