# Calc Expected Value & Variance of Multivar. Func.

• MHB
• TheFallen018
In summary: X is 110 and the variance is $20^2= 400$. The variance of any function of X, such as C, is $a^2$ times the variance of X when you multiply X by a and add another number, a is the number multiplying X. You have $C= 1.9X+ 2.9Y$ so the expected value of C is $1.9(110)+ 2.9(2)= 209+ 5.8= 214.8$ and the variance of C is $1.9^2(400)+ 2.9^2(400)= 361+ 841= TheFallen018 Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. View attachment 8906 Find$E(C)$Find$Var(C)$I tried to get the limits from the normal distributions that were given. If I was doing it right, I had$90\leq X \leq 130 1.9\leq Y \leq 2.9$for X and Y. I think my main problem is that I'm not sure how to get$f(x)$and$f(y)$so that I can use the property$E(aX+bY) = aE(X)+bE(Y)$and$E(x) = \int_{n}^{m}x*f(x)dx$where$n \leq f(x) \leq m$. I think I might be able to figure it out once I can work out what the functions should be, but I'm a little stuck here. Any help would be awesome. Thanks. #### Attachments • Screenshot_53.png 8.1 KB · Views: 67 Last edited: You are told that X is normally distributed with mean 110 and standard deviation 20 (I am assuming that the second number, "$20^2$" is the variance that is the square of the standard deviation.) Whoever gave you this problem clearly expects you to know what the "normal distribution" is! The "f(x)" you want is$\frac{1}{\sqrt{2(\pi)(20^2)}}e^{-\frac{(x- 110)^2}{2(20^2)}} = \frac{1}{20\sqrt{2\pi}}e^{\frac{(x- 110)^2}{800}}\$ and similarly for g(y). (You have "f(x)" and "f(y)" but they are not the same function!)

Look at https://en.wikipedia.org/wiki/Normal_distribution

## 1. What is the expected value of a multivariate function?

The expected value of a multivariate function is the average value that the function takes on over all possible inputs. It is calculated by taking the sum of the product of each input with its respective probability, where the probabilities must sum to 1.

## 2. How is the expected value of a multivariate function calculated?

The expected value of a multivariate function is calculated by taking the sum of the product of each input with its respective probability, where the probabilities must sum to 1. This can also be represented mathematically as E[f(x,y)] = ∑∑ f(x,y)·P(x,y).

## 3. What is the variance of a multivariate function?

The variance of a multivariate function measures how much the function values deviate from the expected value. It is calculated by taking the sum of the squared differences between each function value and the expected value, weighted by their respective probabilities.

## 4. How is the variance of a multivariate function calculated?

The variance of a multivariate function is calculated by taking the sum of the squared differences between each function value and the expected value, weighted by their respective probabilities. This can also be represented mathematically as Var[f(x,y)] = ∑∑ (f(x,y) - E[f(x,y)])^2 · P(x,y).

## 5. What is the significance of calculating the expected value and variance of a multivariate function?

Calculating the expected value and variance of a multivariate function allows us to understand the behavior and variability of the function over all possible inputs. This information can be useful in making predictions and decisions based on the function, as well as in evaluating the effectiveness of different models or algorithms. Additionally, these calculations are fundamental in many statistical and machine learning techniques.

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