Sequence of random variables

In summary, the conversation is about solving an exercise involving a sequence of random variables following a uniform distribution and a random variable defined as the first point at which the sequence stops decreasing. The conversation covers three parts of the exercise, with the third part being already solved. For the first part, the use of induction is suggested and some simple cases are recommended to understand the general case. The summary also points out a mistake in the expression given for the first part and clarifies the use of "^" signs and parentheses. The second part is also briefly discussed, but there is confusion about the notation used for "N pair".
  • #1
Dassinia
144
0
Hi, I'm trying to solve this exercise but I really don't know how
1. Homework Statement

Let X1, X2,.. be a sequence of iid random variables following a uniform distribution on (0,1). Define the random variable N≥2 as the first point in which the sequence (X1,X2,...) stops decreasing. i.e If N=n :
X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifX2[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gif...[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifXn-1<Xn

For 0[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gif1 show that

1. P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N=n)=tn-1/(n-1)! - tn/(n)!
2.P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N pair)=1-exp(-t) use series expansion of exp(t)+exp(-t) et exp(t)-exp(-t)
3. E[N]=e

Homework Equations

The Attempt at a Solution


I solved the third one
For the first one, I think that we have to use induction proof, but I don't see how to do that here ?
Thanks
 
Last edited by a moderator:
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  • #2
Dassinia said:
Hi, I'm trying to solve this exercise but I really don't know how
1. Homework Statement

Let X1, X2,.. be a sequence of iid random variables following a uniform distribution on (0,1). Define the random variable N≥2 as the first point in which the sequence (X1,X2,...) stops decreasing. i.e If N=n :
X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifX2[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gif...[PLAIN]http://www.ilemaths.net/img/smb-bleu/supegal.gifXn-1<Xn

For 0[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gif1 show that

1. P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N=n)=tn-1/(n-1)! - tn/(n)!
2.P(X1[PLAIN]http://www.ilemaths.net/img/smb-bleu/infegal.gift,N pair)=1-exp(-t) use series expansion of exp(t)+exp(-t) et exp(t)-exp(-t)
3. E[N]=e

Homework Equations

The Attempt at a Solution


I solved the third one
For the first one, I think that we have to use induction proof, but I don't see how to do that here ?
Thanks

For the first one, try out some simple cases first: do it for n = 2, n = 3, etc. You will soon see how to do the general case.

What you wrote for 1) is wrong: you say
[tex] P(X_1 \leq t, N=n) = tn - \frac{1}{(n-1)!} - \frac{tn}{n!}[/tex]
when your expression is parsed using standard mathematical rules. I suspect you might have meant
[tex] \frac{t^{n-1}}{(n-1)!} - \frac{t^n}{n!} [/tex]
In that case, you MUST use "^" signs and parentheses: tn means ##t \times n##, but t^n means ##t^n##. Similarly, t^n-1 means ##t^n - 1##, but t^(n-1) means ##t^{n-1}##.

Finally, I do not understand what "X1 ≤ t, N pair" means---that is, what is "N pair"?
 
Last edited by a moderator:

Related to Sequence of random variables

What is a sequence of random variables?

A sequence of random variables is a set of random variables that are indexed by a specific parameter, such as time or position. Each random variable in the sequence represents a different outcome of a random process.

What is the difference between a sequence of random variables and a random variable?

A random variable is a single variable that can take on different values with certain probabilities. A sequence of random variables is a collection of random variables that are related to each other and are often used to model a time-dependent or spatially dependent process.

What is the importance of studying the sequence of random variables?

Studying the sequence of random variables allows us to better understand and model real-life processes that are subject to randomness. It also helps us make predictions and decisions based on the patterns and behaviors of the sequence.

What are some common examples of a sequence of random variables?

Some common examples of a sequence of random variables include stock prices over time, daily temperatures in a specific location, and the number of customers entering a store each hour.

What is the role of probability in a sequence of random variables?

Probability plays a crucial role in a sequence of random variables as it allows us to quantify the likelihood of each outcome in the sequence. It also helps us analyze and make predictions about the future behavior of the sequence.

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