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Category Archives: Mean and Variance
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.
- What is the distribution of number of rounds played in a single game?
- What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
- 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.
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.
- In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
- What is \(\mathbb{E}(X_{160})\)?
- What is \(Var(X_{160})\)?
- Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).
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?
Variance of binomial vs. hypergeometric
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.
- What is the difference between the variance of \(X\) and the variance of \(Y\) ?
- For what values of \(N,n,r\) is the variance the largest ?
Estimating differences of independent draws
- Show that if \(X\) and \(Y\) are independent random variables, then
\[\mathrm{Var}(X- Y)=\mathrm{Var}(X+Y)\] - 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]
Unions and intersections of events
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 \(B\).
- 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 \(\hat A\cap \hat B = \emptyset\).
- Show that \(\mathbf{P}(A)= \mathbf{P}(\hat A) + \mathbf{P}(C)\) and \(\mathbf{P}(B)= \mathbf{P}(\hat B) + \mathbf{P}(C)\) .
- Show that \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B) – \mathbf{P}(A \cap B)\).
Sums of normals
- 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 \(Z=X+Y\)
- What is the distribution of \(Z\) ?
- What is the mean and variance of \(Z\) ?
- (**) If we now assume that they are not independent, but still normal as described above, what can you say ?
Standardized Random Variables
Consider a random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). Define a new random variable \(Y\) by
\[Y=\frac{X-\mu}{\sigma}\,.\]
- Show that \(Y\) has mean 0 and variance 1.
- Show that if \(a \) is some number \[\mathbf{P}( Y > a) = \mathbf{P}( X > \mu + a\sigma )\]
