Mathematics

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.
7247 questions
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Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
leava_sinus
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5 answers

Intuition behind using complementary CDF to compute expectation for nonnegative random variables

I've read the proof for why $\int_0^\infty P(X >x)dx=E[X]$ for nonnegative random variables (located here ) and understand its mechanics, but I'm having trouble understanding the intuition behind this formula or why it should be the case at all.…
bouma
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Why does the median minimize $E(|X-c|)$?

Suppose $X$ is a real-valued random variable and let $P_X$ denote the distribution of $X$. Then $$ E(|X-c|) = \int_\mathbb{R} |x-c| dP_X(x). $$ The medians of $X$ are defined as any number $m \in \mathbb{R}$ such that $P(X \leq m) \geq \frac{1}{2}$…
Tim
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66
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3 answers

Expectation of Minimum of $n$ i.i.d. uniform random variables.

$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)? I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to…
jet
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11 answers

Striking applications of linearity of expectation

Linearity of expectation is a very simple and "obvious" statement, but has many non-trivial applications, e.g., to analyze randomized algorithms (for instance, the coupon collector's problem ), or in some proofs where dealing with non-independent…
Clement C.
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60
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3 answers

Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$

Let $X$ be a non-negative random variable and $F_{X}$ the corresponding CDF. Show, $$E(X) = \int_0^\infty (1-F_X (t)) \, dt$$ when $X$ has : a) a discrete distribution, b) a continuous distribution. I assumed that for the case of a continuous…
Jon Gan
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7 answers

What is the difference between Average and Expected value?

Question : What is the difference between Average and Expected value? I have been going through the definition of expected value on Wikipedia beneath all that jargon it seems that the expected value of a distribution is the average value of the…
user2340452
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45
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3 answers

A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?

A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random points and $7$ clusters, with an average cluster size…
Dan
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8 answers

100 prisoners and a lightbulb

100 prisoners are imprisoned in solitary cells. Each cell is windowless and soundproof. There's a central living room with one light bulb; the bulb is initially off. No prisoner can see the light bulb from his or her own cell. Each day, the warden…
steadyboy
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42
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4 answers

Free throw interview question

I recently had an interview question that posed the following... Suppose you are shooting free throws and each shot has a 60% chance of going in (there are no "learning" or "depreciation" effects, all have the some probability no matter how many…
sedavidw
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Pascal gets drunk and builds his triangle. Half the time he writes $0$ instead of adding the two numbers above. What is the average of the numbers?

One day Pascal has a little too much to drink, then sits down to build his famous triangle. He writes $1$ 's going down the sides, no problem. Then he starts the arduous task of filling in the triangle. Each number is supposed to be the sum of the…
Dan
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6 answers

An ant walk on the edges of a cube where it can select any of the 3 adjoining vertices with equal probability.

There is a cube and an ant is performing a random walk on the edges where it can select any of the 3 adjoining vertices with equal probability. What is the expected number of steps it needs till it reaches the diagonally opposite vertex?
Neel
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36
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11 answers

When to stop rolling a die in a game where 6 loses everything

You play a game using a standard $6$ -sided die. You start with $0$ points. Before every roll, you decide whether you want to continue the game or end it and keep your points. After each roll, if you rolled $6$ , then you lose everything and the…
Kulawy Krul
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32
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On average, how many times must a circular pizza be randomly cut, to get a piece with no curved edge?

On a circular pizza, we make a random straight cut by choosing two uniformly random points on the perimeter and cutting through them. On average, how many times must the pizza be randomly cut, to get a piece with no curved edge? (In other words, on…
Dan
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32
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'Trace trick' for expectations of quadratic forms

I am trying to understand the proof for the Kullback-Leibler divergence between two multivariate normal distributions . On the way, a sort of trace trick is applied for the expectation of the quadratic form $$E[ (x-\mu)^T \Sigma^{-1} (x-\mu) ]=…
tomka
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