Confused by this probability question

[das1]

I came across this problem and I'm wondering if anyone can tell me what it means/how to do it:
"Given the formula below to model, what is the expected value of rolling two dice simultaneously?
$$E(X)= \sum_{i=1}^{\infty}x_i f(x_i)."$$

I've never seen this notation before; how does it work?

 

[Fermat1]

I came across this problem and I'm wondering if anyone can tell me what it means/how to do it:
"Given the formula below to model, what is the expected value of rolling two dice simultaneously?
E[X] = *the sum from 1 to infinity* of x_i f (x_i)"

I've never seen this notation before; how does it work?

You are dealing with a discrete random variable (the total sum on two dice is an integer) so the expectation won't be an integral, it will be a sum. Let $X$ be the sum of the two dice. $f_{X}(x)$ is a common notation for the density fuction. For discrete Random variables, as in this case we have $f_{X}(x)=P(X=x)$. $X$ can take values 1 to 12. The expectation is $1P(X=1)+2P(X=2)+..+12P(X=12)$

 

[das1]

Fermat, you're a legend, thank you.
One thing, shouldn't it be 2 to 12 because that's the lowest you can get when rolling 2 dice simultaneously?

 

[Jameson]

Fermat, you're a legend, thank you.
One thing, shouldn't it be 2 to 12 because that's the lowest you can get when rolling 2 dice simultaneously?

Good question! Normally you write all non-zero terms in the sum but you can actually use this formula for all discrete numbers. Why? Let's look at when the sum equals $-2$ for example. This part of the expected value sum would be $-2 \cdot P[X=-2]$. The probability the sum equals a negative number though is obviously 0, so this term becomes $-2 \cdot 0=0$. All other terms outside of the range of possible sums will also drop to zero, thus we usually only include the non-zero terms when writing out the work.

So yes, it would make more sense maybe to write $2P(X=2)+3P[X=3]..+12P(X=12)$ but $1P(X=1)+2P(X=2)+..+12P(X=12)$ isn't incorrect because $1P[X=1]=0$ :)

 

[CellCoree]

I can help clarify this notation for you. The formula given is the mathematical representation for calculating the expected value of a random variable, in this case, the sum of two dice rolls. The notation may seem unfamiliar, but it is a standard way of writing mathematical expressions.

The "E(X)" represents the expected value of the random variable X, which is the sum of the two dice rolls. The summation symbol, Σ, indicates that we are adding up all the possible outcomes of the random variable. The "i=1" below the summation symbol means that we are starting at the first possible outcome, which is rolling a 1 on both dice. The "∞" above the summation symbol indicates that we are considering all possible outcomes, up to an infinite number.

The "x_i" represents each possible outcome, in this case, the sum of two dice rolls. So, for example, x_1 would be the outcome of rolling a 1 on both dice, x_2 would be the outcome of rolling a 2 on both dice, and so on. The "f(x_i)" represents the probability of each outcome occurring. So, for example, f(x_1) would be the probability of rolling a 1 on both dice, f(x_2) would be the probability of rolling a 2 on both dice, and so on.

To calculate the expected value using this formula, you would need to know the probability of each possible outcome. Once you have that information, you can plug it into the formula and solve for the expected value. I hope this helps clarify the notation and how to use it to solve the problem.