Theoretical Stats Problem

In summary: X1 and X2 are odd. Using the same logic as in part (a), the probability that both X1 and X2 are even is E/n, where E is the total number of even values of X1 and X2. The probability that both X1 and X2 are odd is O/n, where O is the total number of odd values of X1 and X2. To find the total number of even values of X1 and X2, we need to consider the possible combinations of even and odd values for X1 and X2. For example, if X1=2 and X2=4, then X1+X2=6, which is even. If X1=4
  • #1
uva123
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Homework Statement



for each value of p>1

c(p)= 1/xp

(the n=1 under the sumation symbol should be x=1)

Suppose that the random variable X has a discrete distribution with the following p.f.:

f(x)= 1/[c(p)xp] for x=1,2,...


(a) For each fixed positive integer n, determine the probability that \ will be divisible by n;

(b) Determine the probability that X will be odd.

(c) Suppose that X1 and X2 are independent random variables, each of which has the p.f.
above. Determine the probability that X1+X2 will be even
and the probability that X1+X2 will be odd.


Homework Equations



-c(p) is a convergent p series

3. The Attempt at a Solution

-I don't even know where to start! please help!
-does f(x) reduce to 1? if so then x=1 so x only divisible by n is n=1 and the probability x is odd would be 1 because x is an odd #
 
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  • #2


Hello! Let's break this down step by step to make it easier to understand.

(a) For each fixed positive integer n, determine the probability that X will be divisible by n.

To determine the probability that X will be divisible by n, we need to find the values of x that are divisible by n. In this case, we know that x=1,2,... so we need to find the values of x that are divisible by n within this range. This can be done by setting up an equation:

x=kn, where k is a positive integer.

So, for example, if n=2, then the values of x that are divisible by 2 would be 2, 4, 6, 8, etc.

To find the probability, we need to find the total number of values of x that are divisible by n (let's call this X) and divide it by the total number of possible values of x (which is the same as the total number of values of x, which is n).

So the probability that X will be divisible by n is X/n.

(b) Determine the probability that X will be odd.

To determine the probability that X will be odd, we need to find the values of x that are odd within the range of x=1,2,... We know that an odd number is any number that cannot be divided by 2 without a remainder. So, the values of x that are odd are 1, 3, 5, 7, etc.

As mentioned in part (a), to find the probability, we need to find the total number of values of x that are odd (let's call this O) and divide it by the total number of possible values of x (which is the same as the total number of values of x, which is n).

So the probability that X will be odd is O/n.

(c) Suppose that X1 and X2 are independent random variables, each of which has the p.f. above. Determine the probability that X1+X2 will be even and the probability that X1+X2 will be odd.

To determine the probability that X1+X2 will be even, we need to find the values of X1+X2 that are even.

For X1+X2 to be even, either both X1 and X2 are even, or
 

1. What is the difference between theoretical and applied statistics?

Theoretical statistics involves developing mathematical models and theories to understand and explain data, while applied statistics involves using these models to analyze real-world data and make predictions or decisions.

2. How is probability used in theoretical statistics?

Probability is used to quantify the likelihood of different outcomes in a theoretical statistical model. It allows us to make predictions about the behavior of a system or population based on the underlying assumptions and parameters of the model.

3. What are some common applications of theoretical statistics?

Theoretical statistics is used in a variety of fields, including economics, biology, psychology, and engineering. It can be used to analyze and predict trends in financial markets, evaluate the effectiveness of medical treatments, and understand the behavior of complex systems.

4. How do you validate a theoretical statistical model?

There are several ways to validate a theoretical statistical model, including comparing it to real-world data, conducting hypothesis testing, and performing sensitivity analysis to see how changes in the model's assumptions impact the results.

5. What are some common challenges in working with theoretical statistics?

Some common challenges in theoretical statistics include identifying appropriate models for a given data set, dealing with complex or incomplete data, and interpreting the results of a model in a meaningful way. Additionally, theoretical models may not always accurately reflect real-world data, leading to limitations in their application.

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