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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.
7219 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. Does…
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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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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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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6 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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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
  • 35,053
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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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35
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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 game…
Kulawy Krul
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1 answer

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,…
Dan
  • 35,053
32
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2 answers

'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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