Propability Density (I'm stuck need assistance)

In summary, X is a stochastic variable with a distribution function F_X given by P(X<=x) = F_X(x), where the values are 0 for x>0, (1-e^-x)/(1-e^-1) for x in [0,1], and 1 for x>=1. The problem is to show that X is absolutely continuous and calculate the probability density function f_x. The probability density is found by differentiating the distribution function, which gives F'_{(X)}(x) = (e^(1-x))/(e-1). The next step is to use this to find the probability density. To show that X is absolutely continuous, it is enough to show that the distribution function is different
  • #1
Mathman23
254
0
Hi
I have been given this following problem:
X is a stochastic variable with the distribution Function F_X which is given by:
[tex]
P(X \leq x) = F_{X}(x) = \left\{ \begin{array}{ll}
0 & \textrm{if} \ x>0 \\
\frac{{1- e^{-x}}}{{1 - e^{-1}}}& \textrm{if} \ x \in [0,1]\\
1 & \textrm{if} \ x \geq 1\\
\end{array} \right.
[/tex]
Now I'm supposed to show that X is absolutely continuous and then next calculate the propability density f_x.
I now then dealing with the propability density is found by
[tex]F'_{(X)}(x) = \frac{e^{1-x}}{e-1}[/tex]
But what is the next step from here which will allow me to find the propability density?
Secondly how do I go about showing that X is absolutly continuous ??
Sincerely and Best Regards
Fred
 
Last edited:
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  • #2
Hello again,

My own solution:

Since [tex]F_{X} (x) [/tex] is differentiable, thereby according to the differention its also continious (but how is it absolutely continious??))

The density is therefore

[tex]p(x) = \frac{e^{1-|x|}}{e-1} [/tex]

Could anyone please inform me if I'm on the right course?

Best Regards,

Fred

Mathman23 said:
Hi
I have been given this following problem:
X is a stochastic variable with the distribution Function F_X which is given by:
[tex]
P(X \leq x) = F_{X}(x) = \left\{ \begin{array}{ll}
0 & \textrm{if} \ x>0 \\
\frac{{1- e^{-x}}}{{1 - e^{-1}}}& \textrm{if} \ x \in [0,1]\\
1 & \textrm{if} \ x \geq 1\\
\end{array} \right.
[/tex]
Now I'm supposed to show that X is absolutely continuous and then next calculate the propability density f_x.
I now then dealing with the propability density is found by
[tex]F'_{(X)}(x) = \frac{e^{1-x}}{e-1}[/tex]
But what is the next step from here which will allow me to find the propability density?
Secondly how do I go about showing that X is absolutly continuous ??
Sincerely and Best Regards
Fred
 
Last edited:

1. What is probability density?

Probability density is a concept in statistics that describes the likelihood of a continuous random variable falling within a specific range of values. It is represented by a probability density function (PDF) and can be used to calculate the probability of obtaining a particular value within a continuous distribution.

2. How is probability density different from probability?

Probability refers to the likelihood of a specific outcome occurring, while probability density measures the likelihood of a range of outcomes occurring. In other words, probability density provides a more detailed and continuous description of the likelihood of obtaining different values compared to a discrete probability distribution.

3. What are the units of probability density?

The units of probability density depend on the specific problem and the units of the random variable being measured. For example, if the random variable is time, the units of probability density would be "per second". If the random variable is distance, the units would be "per meter". It is important to include the appropriate units when interpreting probability density.

4. How is probability density related to the normal distribution?

The normal distribution is a specific type of probability density function that is commonly used in statistics. It is characterized by a bell-shaped curve and is often used to model real-world phenomena. However, there are many other types of probability density functions that can be used depending on the problem being studied.

5. How can probability density be used in scientific research?

Probability density is a fundamental concept in statistics and is widely used in scientific research. It can be used to analyze and interpret data, make predictions, and test hypotheses. For example, in psychology, probability density can be used to analyze the distribution of scores on a standardized test and determine the likelihood of obtaining a certain score. In physics, probability density can be used to describe the behavior of particles in a quantum system.

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