Category Archives: Mean and Variance

Rock Paper Scissors

Posted on October 29, 2022 by | Comments Off on Rock Paper Scissors

A game of rock paper scissors consists of several rounds (players continue to play rounds until one player wins). In one round of rock paper scissors, two players each choose one of three options (rock, paper, or scissors). If they choose two different options, the game ends (rock beats scissors, scissors beat paper, and paper beats rock). If both players choose the same option, the game continues for another round. Assume each player chooses rock, paper or scissors uniformly at random and independently.

  1. What is the distribution of number of rounds played in a single game?
  2. What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
  3. Let \(n\) be the number of games played. How big must \(n\) be to ensure at least 100 rounds are played with 90% probability? Use an appropriate approximation to estimate.

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Posted in Geometric Distribution, Mean and Variance, Normal/CLT approximation

Ant crawls along a number line

Posted on October 29, 2022 by | Comments Off on Ant crawls along a number line

An ant is crawling on a number line. The ant starts out at position \(0\). Every second the ant either

  • Moves to the right 1 unit, with probability 1/2,
  • Moves to the left 1 unit, with probability 1/4, or
  • Stays at its current location, with probability 1/4

The ant’s movement during a particular second is independent of the ant’s previous movements. Let \(X_{160}\) be the ant’s location after 160 seconds.

  1. In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
  2. What is \(\mathbb{E}(X_{160})\)?
  3. What is \(Var(X_{160})\)?
  4. Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).

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Posted in Mean and Variance, Multinomial, Normal/CLT approximation

Mean and Mode of Binomial

Posted on October 29, 2022 by | Comments Off on Mean and Mode of Binomial

Consider a binomial\((10,p)\) distribution. If \(p\) is chosen uniformly at random from the interval \((0,1)\), what is the likelihood that the most likely number of the binomial distribution will be less than the mean of the binomial distribution?

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Posted in Binomial, Mean and Variance, Uniform

Variance of binomial vs. hypergeometric

Posted on September 20, 2012 by | Leave a comment

Given \(N\) balls of which \(r\) of them are red and the rest are green. Denote \(X\) as the number of red balls drawn when sampling with replacement and \(Y\) as the number of red balls drawn when sampling without replacement.

  1. What is the difference between the variance of \(X\) and the variance of \(Y\) ?
  2. For what values of \(N,n,r\) is the variance the largest ?

 

 

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Posted in Binomial, hypergeometric distribution, Mean and Variance

Estimating differences of independent draws

Posted on August 17, 2012 by | Comments Off on Estimating differences of independent draws
  1. Show that if \(X\) and \(Y\) are independent random variables, then
    \[\mathrm{Var}(X- Y)=\mathrm{Var}(X+Y)\]
  2. Let \(D_1\) and \(D_2\) represents two draws at random with replacement from a population, with \(\mathbf{E}D_1=10\) and \(\mathbf{SD}(D_1)_1=2\). Find a number \(c\) so that
    \[\mathbf{P}(|D_1 -D_2| < c) \geq .99\]

From [Pittman p. 203, #15]

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Posted in Bounding Probabilies, Mean and Variance

Unions and intersections of events

Posted on August 13, 2012 by | Leave a comment

Let \(A\) and \(B\) be any two events. Define the new events \(C\), \(\hat A\), and \(\hat B\) by   \(C=A\cap B\), \(\hat A=A \cap B^c\), and \(\hat B = B \cap A^c\) where \(A^c\) is the compliment of \(A\) and \(B^c\) is the compliment of \(B\).

  1. Argue  that \(A \cup B = \hat A \cup \hat B \cup C\) and that all three sets are mutually  disjoint. i.e. \(\hat A\cap C = \emptyset\), \(\hat B\cap C = \emptyset\), and \(\hat A\cap \hat B = \emptyset\).
  2. Show that \(\mathbf{P}(A)= \mathbf{P}(\hat A) + \mathbf{P}(C)\) and \(\mathbf{P}(B)= \mathbf{P}(\hat B) + \mathbf{P}(C)\) .
  3. Show that \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B)  – \mathbf{P}(A \cap B)\).

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Posted in Basic probability, Manipulating Events, Mean and Variance

Sums of normals

Posted on July 25, 2012 by | Leave a comment
  • Consider a normal random variable \(X\) with mean \(\mu_1\) and standard deviation \(\sigma_1\)
  • Consider a normal random variable \(Y\) with mean \(\mu_2\) and standard deviation \(\sigma_2\).

Assume that \(X\) and \(Y\) are independent and define \(Z=X+Y\)

  1. What is the distribution of \(Z\) ?
  2. What is the mean and variance of \(Z\) ?
  3. (**) If we now assume that they are not independent, but still normal as described above, what can you say ?

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Posted in Basic probability, Mean and Variance

Tagged

Standardized Random Variables

Posted on July 25, 2012 by | Leave a comment

Consider a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). Define a new random variable \(Y\) by
\[Y=\frac{X-\mu}{\sigma}\,.\]

  1. Show that \(Y\) has mean 0 and variance 1.
  2. Show that if \(a \) is some number  \[\mathbf{P}( Y > a) = \mathbf{P}( X > \mu + a\sigma )\]

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Posted in Basic probability, Mean and Variance