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Questions tagged [expected-value]

Questions about the expected value of a random variable.

The average value of a randomly chosen quantity is its expectation or expected value. For example, the expected value of the number you get when you roll a fair 6-sided dice is 3.5.

In general, if $X$ is a random variable defined on a probability $(\Omega, \Sigma, P)$, then the expected value of $X$, denoted by $E[X], \langle X \rangle,$ or $\bar{X}$ is defined as the Lebegue integral

$$E[X]= \int_{\Omega} X(\omega) dP(\omega)$$

The expected value is often the first and most important thing you want to know about a random variable. For example, in a betting game, the best strategy is often the one that maximizes the expected value of the amount you win.

This tag is for questions about:

  • Computing the expected value in a specific situation.
  • Understanding the properties of expected values, such as Markov's inequality or linearity of expectation.
  • Proving theorems about the expected value of abstract random variables.
  • Understanding what the expected value means and what it tells you about a random variable.
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how many draws does Nate need until the expected number of diamonds he picks is 1

A chest contains 4 rubies, 8 emeralds, 3 diamonds, and 3 sapphires. Nate will draw jewels one at a time. After each draw, he puts the jewels back into the chest. Nate will go free if he picks some diamonds. How many draws does Nate need until the…
serendipity0217
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Probability that the blue disk is in the center after $n$ moves

Jimmy has two red disks and one blue disk, and he places them in a row such that the blue disk is in the center. Every move, Jimmy switches one of the outer disks with the center disk. Find the probability that the blue disk is in the center after…
questionasker
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Marginal mass function involving Y=r(X)

Let $X \sim \operatorname{Unif}(0,1)$. Let $Y=r(X)=e^{X}$. Then, $$ \mathbb{E}(Y)=\int_{0}^{1} e^{x} f(x) d x=\int_{0}^{1} e^{x} d x=e-1 $$ Alternatively, you could find $f_{Y}(y)$ which turns out to be $f_{Y}(y)=1 / y$ for $1
critterTI
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Fast way to calculate new expected value of card game

Let's say there are 13 cards, labeled from 1 to 13, and 5 players, each who get a card at random. Every player can see their own card, and that is all. The players are trying to estimate the sum of the 8 remaining cards (say the remaining cards are…
Eric Aldrin
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Finding $E[Y|X]$ by finding $E[Y | X = x]$ and plugging in $X$ for $x$

I read that if we're conditioning on a random variable ($X$ in our case), then to find $E[Y | X]$, we can simply first find $E[Y | X = x]$, then from this expression, we plug in $X$ everywhere we see $x$, and that gives us $E[Y | X]$. This seems…
student010101
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Expected value of coin flips using indicator variables

I'm wondering how to go about solving this problem. Flip a coin five times. Let $X_i$ be the indicator variable ($X_i = 1$) if the $i$th flip and $i+1$ flips are the same where $1\leq i \leq 4$. What is $E[X_i]$? I'm thinking $X$ is 4 which is the…
yololo
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Does the expected value of the sum of components of a function of a random variable equal the sum of their expected values?

Assuming $f:\mathbb R^N \to \mathbb R^N$ is a differentiable function, and $X \in \mathbb R^N$ a random variable with expectation $\mu$ and finite co-variance matrix $\Sigma$. Does the following hold true, given the linearity of…
PedanticCat
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If $E[|X|^k]$ exists, does $E[|X|^m]$ exist for $0 \leq m \leq k$?

The question is: If $E[|X|^k]$ exists, does $E[|X|^m]$ exist for $0 \leq m \leq k$? If so, can you prove it? Here $X$ is just a generic random variable. I don't know how to solve this problem. I thought about approaching it by first defining $g(x,…
user5965026
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When I'm calculating the expected value of a joint pdf, does it matter whether I integrate with respect to $y$ or $x$ first?

My book says the following: If $X$ and $Y$ are continuous random variables with joint density $f(x, y)$, and if $g$ is any function of the two variables $X$ and $Y$, then: $$E(g(X,Y)) = \int_{-\infty}^\infty\int_{-\infty}^\infty…
user908519
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Expected number of seconds for a monkey to type "greek symbol" on a keyboard.

A monkey types letters from an alphabet plus the space bar with equal probability. He types 1 character per second. What is the expected number of seconds it will take for the monkey to type the phrase inside the quotes "greek symbol" ? The problem…
user5965026
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Expected number of tosses to get either $n$ heads or $n$ tails in a row?

I have solved the problem for the expected number of tosses to get $n$ heads in a row, which is $2^{n+1} - 2$. To get $n$ heads OR $n$ tails, I think the problem is substantially more difficult, and I can't figure out a way to generalize this for…
user5965026
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Find $E[(x+2)^2]$ Given $E[x]$

Find $ E[(x+2)^2] $ given that $ E[x] = 5, Var(X) = 2 $. I'm not sure if I'm making this more complicated than it should be. However, this is what I did. $$ E[(x+2)^2] = E[x^2+2x+4] = E[x^2]+E[2x]+4 = E[x^2]+2E[x]+4 = 25+10+4 = 39. $$ I'm unsure…
Jones
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How to compute the E(x) given its cdf function?

How could I compute the expected value E(x) given that its cdf is, $F(x) = \dfrac{\exp(x)}{1+\exp(x)}$. I know the method to derive E(x), as in this post. I still find it very hard to derive the result. Thanks.
Carl
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How to find an expected value $E[X]$ if you're given the probabilities $P(X < 3) = \frac{1}{3}$ and $P(X \geq 6) = \frac{1}{6}$?

If I know the probabilities of a non-negative random variable constraints $P(X < 3) = \frac{1}{3}$ and $P(X \geq 6) = \frac{1}{6}$, how then I find all possible expected values E[X]? I tried to use Markov Inequality for the second probability to…
chroniclesofanoctupus
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Expected value of dice sum

We roll a dice. If we get 6, then we roll again. What's the expected value of sum of rolls? Let $X$ be the sum. In this game we can only obtain sum in form $6k + x$ where $k \in \mathbb{Z}$ and $x \in \{1,2,3,4,5\}$. If $n$ is in this form then…
Nerwena
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