Hi, can someone please clarify my basic confusion.Let's say if I have two hypothetical random independent variables X and Y like : X : 1,2,3 Y : 4,5,6 Now if I have to combine these two variables what will be resultant output X+Y? Will it be {1,2,3,4,5,6} Or {5,7,9} ?

You can't really add the random variables themselves but can add their means and std dev. You would get mean:7 StdDev.:1.1547

if you like maths so much name every number

If we define a description of a number as a finite string of symbols that uses a finite alphabet of symbols, then there are only countably many descriptions. However, there are uncountably many real numbers. So almost all real numbers are indescribable!

prove that the arithmetic mean of the sum of two or more variables is equal to the sum of their mean.

Math definition of expected value for a continuous RV X is E(X)=integral(x*p(x)). Therefore definition of expected value of sum of two RVs X,Y is E(X+Y)=integral(x*p(x)+y*p(y)). Integral is a linear operator, so integral(x*p(x)+y*p(y))=integral(x*p(x))+integral(y*p(y)) which is E(X)+E(Y). Same is true of discrete RVs, they're defined as sum rather than integral, sum is likewise a linear operator.

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Course: AP®︎/College Statistics > Unit 8

Lesson 4: Combining random variables

Mean of sum and difference of random variables

Name: Mean of sum and difference of random variables
Uploaded: 2017-08-11T01:49:23Z
Description: Mean of sum and difference of random variables

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Mean of sum and difference of random variables.

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rautchetan2993
Posted 6 years ago. Direct link to rautchetan2993's post “Hi, can someone please cl...”
Hi, can someone please clarify my basic confusion.Let's say if I have two hypothetical random independent variables X and Y like :

X : 1,2,3
Y : 4,5,6

Now if I have to combine these two variables what will be resultant output X+Y?
Will it be {1,2,3,4,5,6}
Or {5,7,9} ?
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(11 votes)
Answer
- Saivishnu Tulugu
  Posted 5 years ago. Direct link to Saivishnu Tulugu's post “You can't really add the ...”
  You can't really add the random variables themselves but can add their means and std dev.
  You would get mean:7
  StdDev.:1.1547
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  (5 votes)
Rishabh Chopra
Posted 5 years ago. Direct link to Rishabh Chopra's post “Where is the proof of thi...”
Where is the proof of this? Please help with the link?
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(6 votes)
Answer
Ping
Posted 4 years ago. Direct link to Ping's post “if you like maths so much...”
if you like maths so much name every number
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(2 votes)
Answer
- Ian Pulizzotto
  Posted 4 years ago. Direct link to Ian Pulizzotto's post “If we define a descriptio...”
  If we define a description of a number as a finite string of symbols that uses a finite alphabet of symbols, then there are only countably many descriptions. However, there are uncountably many real numbers. So almost all real numbers are indescribable!
  Comment on Ian Pulizzotto's post “If we define a descriptio...”
  (3 votes)
Janeyh
Posted 3 years ago. Direct link to Janeyh's post “I still don't know whats ...”
I still don't know whats a sum
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(1 vote)
Answer
- fumperdinq
  Posted 4 months ago. Direct link to fumperdinq's post “the sum in a layperson's ...”
  the sum in a layperson's term would be the result when you add them together.
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  (1 vote)
leena kiyumi
Posted 4 years ago. Direct link to leena kiyumi's post “so is there a difference ...”
so is there a difference between an expected value and a mean? like can the mean be decimals like 5.6 but you can expect to see 5.6 cats, so would the expected value be 6 cause you would round this number upwards? or would it be 5 cause that would give you a more accurate expected value? or am i thinking too much and the expected value would actually just be equal to the mean and therefor be 5.6 ?
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(1 vote)
Answer
- Jerry Nilsson
  Posted 4 years ago. Direct link to Jerry Nilsson's post “Expected value _is_ the a...”
  Expected value is the average value.
  
  So, seeing 5.6 cats could very well be the expected value, even though it's definitely not the expected outcome of any given trial.
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  (1 vote)
moose
Posted 9 months ago. Direct link to moose's post “Does anyone know where I ...”
Does anyone know where I could find the proof video of this?
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(1 vote)
Answer
- mahmoudsamawal
  Posted 8 months ago. Direct link to mahmoudsamawal's post “I'm also stuck here and h...”
  I'm also stuck here and how do we actually combine two random variables like literally how do we construct the resultant distribution?, what did you do?
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  (1 vote)
Mez Cooper
Posted 5 years ago. Direct link to Mez Cooper's post “The expected value is alw...”
The expected value is always a mean? A sample mean? A population mean?

Expected value is the sum of multiplying probabilities with their respective events? Weird.
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(0 votes)
Answer
- daniella
  Posted a month ago. Direct link to daniella's post “The expected value, denot...”
  The expected value, denoted as E[X], is a concept in probability theory representing the theoretical average outcome of a random variable over a large number of trials or occurrences. It's calculated by multiplying each possible value of the random variable by its corresponding probability of occurrence and summing up these products. The expected value provides insight into the central tendency or average behavior of the random variable. It's essential to understand that while the expected value is often referred to as the mean, it's distinct from sample means or population means commonly used in statistics, which are calculated from observed data rather than theoretical probabilities.
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  (1 vote)
halimajaman18
Posted 5 years ago. Direct link to halimajaman18's post “prove that the arithmetic...”
prove that the arithmetic mean of the sum of two or more variables is equal to the sum of their mean.
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(0 votes)
Answer
- sayhitoteebee
  Posted 4 years ago. Direct link to sayhitoteebee's post “Math definition of expect...”
  Math definition of expected value for a continuous RV X is E(X)=integral(x*p(x)). Therefore definition of expected value of sum of two RVs X,Y is E(X+Y)=integral(x*p(x)+y*p(y)). Integral is a linear operator, so integral(x*p(x)+y*p(y))=integral(x*p(x))+integral(y*p(y)) which is E(X)+E(Y). Same is true of discrete RVs, they're defined as sum rather than integral, sum is likewise a linear operator.
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  (2 votes)

Video transcript

- [Instructor] Let's say that I have a random variable X which is equal to the number of dogs that I see in a day. And random variable Y is equal to the number of cats that I see in a day. Let's say I also know what the mean of each of these random variables are, the expected value. So the expected value of X which I could also denote as the mean of our random variable X let's say I expect to see three dogs a day and similarly for the cats, the expected value of Y is equal to I could also denote that as the mean of Y is going to be equal to and this is just for the sake of (mumbles) let's say I expect to see four cats a day. And pretty much we define how you take the mean of a random variable or the expected value for a random variable. What we're going to think about now is what would be the expected value of X plus Y or other way of saying that the mean of the sum of these two random variables. Well it turns out, and I'm not proving it just yet, that the mean of the sum of random variables is equal to the sum of the means. So this is going to be equal to the mean of random variable X plus the mean of random variable Y. And so in this particular case, if I were to say well what's the expected number of dogs and cats that I would see in a given day. Well I would add these two means, it would be three plus four it would be equal to seven, so in this particular case it is equal to three plus four which is equal to seven. And similarly if I were to ask you the difference if I were to say how many more cats in a given day would I expect to see than dogs, so the expected value of Y minus X. What would that be? Well intuitively you might say well hey if we can add random... If the expected value of the sum is the sum of the expected values, then the expected value or the mean of the difference will be the differences of the means and that is absolutely true. So this is the same thing as the mean of Y minus X which is equal to the mean of Y is going to be equal to the mean of Y minus the mean of X, minus the mean of X. And in this particular case, it would be equal to four minus three, minus three is equal to one. So another way of thinking about this intuitively is I would expect to see on a given day one more cat than dog. The example that I just used this is discrete random variables, on a given day I wouldn't see 2.2 dogs or pi dogs, the expected value itself does not have to be a whole number 'cause you could of course average it over many days. But this same idea that the mean of a sum is the same thing as a sum of means and the mean of a difference of random variables is the same as the difference of the means. In a future video I'll do a proof of this.