Find c for Random Variable: E(X), E(X^2), E(1/X) & Var(X)

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Discussion Overview

The discussion revolves around finding the constant c for a discrete random variable X with a specified probability mass function (pmf) and subsequently calculating the expected values E(X), E(X^2), E(1/X), and the variance Var(X). The scope includes theoretical understanding of pmfs and their properties, as well as practical calculations related to random variables.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents the problem of finding c given the pmf P(X=j) = j/c for j in {1, 2, ..., n}.
  • Another participant suggests that the constant c can be determined from the pmf and that expectations can be calculated directly from definitions.
  • A participant expresses confusion about the variable j and its relation to the pmf, indicating difficulty in starting the problem.
  • There is a discussion about the conditions that valid pmfs must satisfy, including the total probability summing to 1 and individual probabilities being between 0 and 1.
  • One participant attempts to set up the equation for the sum of probabilities, leading to the equation 1/c + 2/c + ... + n/c = 1.
  • Another participant questions the assumption that c could equal infinity, suggesting that n is a finite number and prompting a discussion about the sum of the first n positive integers.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the pmf and the calculations required. There is no consensus on how to proceed with finding c, and confusion remains about the relationship between the variables and the properties of pmfs.

Contextual Notes

Participants highlight the need to consider the finite range of the random variable and the specific conditions that define a valid pmf. The discussion includes unresolved mathematical steps related to summing the series of probabilities.

Firepanda
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a discrete random variable has range space {1, 2, ..., n} and satisfies P(X=j) = j/c for some number c. Find c, and then find E(X), E(X^2), E(1/X) and Var(X).

thanks
 
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What is your solution?

You have the probability mass function (pmf). You can determine the constant c.

The expectations are obtained directly by definition...
 
maverick280857 said:
What is your solution?

You have the probability mass function (pmf). You can determine the constant c.

The expectations are obtained directly by definition...

Thanks for your reply.

Sorry I should have stated before that I don't know where to start on this.

I just looked up on pmf, and I have no examples on a question like this in my notes.

The 'j' confuses me in the question as I don't see how it relates to anything else, so finding c is tricky for me.

Thanks
 
Firepanda said:
Sorry I should have stated before that I don't know where to start on this.

Well, aren't there a few conditions that all valid pmf's are required to satisfy? It would probably be a good idea to review these.

Firepanda said:
The 'j' confuses me in the question as I don't see how it relates to anything else, so finding c is tricky for me.

j is simply a dummy variable. Just shorthand for saying P(X=1) = 1/c, P(X=2) = 2/c, ..., P(X=n) = n/c.

Perhaps this thread should be moved to the homework help section?
 
The conditions I know of pmf are that the total sum of the probabilities from -infinity to infinity is 1, and the probabilities can only take values between 0 and 1.

So do I have to find a j/c which has a sum of the series from -infinity to infinity equal to 1?

Sorry I'm really confused..

ok so I have so far:

1/c + 2/c + 3/c ... +n/c = 1

1 + 2 + 3 +.. + n = c

c = infinity?
 
a pmf is for a discrete random variable.

Do you know what the definition of a discrete RV is?

Otherwise you have the right idea. c doesn't have to equal infinity. what is the sum of n consecutive integers?
 
Firepanda said:
The conditions I know of pmf are that the total sum of the probabilities from -infinity to infinity is 1, and the probabilities can only take values between 0 and 1.

So do I have to find a j/c which has a sum of the series from -infinity to infinity equal to 1?
No, only 1 to n, the number over which your probability distribution is defined.

Sorry I'm really confused..

ok so I have so far:

1/c + 2/c + 3/c ... +n/c = 1

1 + 2 + 3 +.. + n = c

c = infinity?
Why should it be? n is a fixed finite number, not "infinity". Do you know the formula for the sum of the first n positive integers?
 

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