# Independent random variables

• WY
In summary, the conversation discusses the problem of finding the probability that the sample total exceeds 88 when dealing with independent random variables from a normal distribution. The individual providing help mentions that the sum of normal random variables is also normally distributed with a mean equal to the sum of the means and a variance equal to the sum of the variances. They also remind the original question-asker to use the standard deviation instead of the variance when changing variables. The conversation concludes with the original question-asker understanding and getting the correct answer.
WY
Hi
I'm wondering if someone can help me out on this question as to how to go about doing it:
X_1, X_2... X_7 are independent random variables represnting a random sample of size 7 from the normal N(10, 7) distribution. Find to 3 dp probablitity that the sample total exceeds 88.

I tried to standardise this but my numbers don't seem to get me the answer of 0.005. Can someone help me out? Thanks in advance :)

How did try to do it? Remember, the d.f. of the sum of random variables with normal distributions is another normal distribution with a mean that is the sum of the means of the individual variables and a variance that is the sum of the variances of the individual variables. Also remember when changing varibles that what appears in the the normal distribution is $$\frac{(x-\mu)}{\sigma}$$ and not $$\frac{(x-\mu)}{\sigma^2}$$, so use the standard deviation and not the variance when changing variables.

Thanks for the help! when i originally did it i used (88-10)/7 to try and standardise it - giving me a ridiculous number. So with the normal distribution N(10,7) what should I now do with those - I'm still kind of confused...

Remember, the mean is the sum of the means of the X_i, so that's 10+10+10+...=70. The variance is the sum of the variances. Remember to normalize with the standard deviation and not the variance. Once you do that you do get the answer you said you were supposed to.

## 1. What are independent random variables?

Independent random variables are variables that have no influence or effect on each other. This means that the values of one variable do not affect the values of the other variable. They are considered independent if the probability of one variable occurring does not change based on the occurrence of the other variable.

## 2. How are independent random variables different from dependent random variables?

Dependent random variables are variables that have an influence or effect on each other. This means that the values of one variable can affect the values of the other variable. They are considered dependent if the probability of one variable occurring changes based on the occurrence of the other variable.

## 3. What is the importance of independent random variables in statistical analysis?

Independent random variables are important in statistical analysis because they allow for simpler and more accurate calculations of probabilities and statistical measures. When variables are independent, the probability of their joint occurrence is simply the product of their individual probabilities. This makes it easier to analyze and make predictions about complex systems.

## 4. How is independence of random variables determined?

Independence of random variables can be determined by calculating the joint probability of their occurrence and comparing it to the product of their individual probabilities. If the joint probability is equal to the product of the individual probabilities, then the variables are considered independent. Additionally, independence can also be determined by examining the relationship between the variables and determining if one variable has any influence on the other.

## 5. Can two random variables be both independent and dependent?

No, two random variables cannot be both independent and dependent at the same time. They are mutually exclusive concepts. A variable can either have no influence on another variable (independent) or have an influence on another variable (dependent). However, two sets of variables can be independent from each other while being dependent within their respective sets.

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