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Approximating sums of uniform random variables

Suppose \(X_1,X_2,X_3,X_4\) are independent uniform \((0,1)\) and we set \(S_4=X_1+X_2+X_3+X_4\). Use the normal approximation to estimate \(\mathbf{P}( S_4 \geq 3) \).

geometric probability: marginal densities

Find the density of the random variable \(X\) when the pair \( (X,Y) \) is chosen uniformly from the specified region in the plane in each case below.

  1. The diamond with vertices at \( (0,2), (-2,0), (0,-2), (2,0) \).
  2. The triangle with vertices \( (-2,0), (1,0), (0,2) \).

[Pitman p 277, #12]

probability density example

Suppose  \(X\) takes values in\( (0,1) \) and has a density

\[f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\  0 & x \not \in (0,1)\end{cases}\]

for some \(c>0\).

  1. Find \( c \).
  2. Find \(\mathbf{E}(X)\).
  3. Find \(\mathrm{Var}(X) \).

 

Raindrops are falling

Raindrops are falling at an average rate of 30 drops per square inch per minute.

  1. What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
  2. If one draws a circle of radius 2 inches on the ground, what is the chance that 4 or more drops hits inside the circle over a two-minute period?
  3. If each drop is a big drop with probability 2/3 and a small drop with probability 1/3, independent of the other drops, what is the chance that during 10 seconds a particular square inch gets hit by precisely four big drops and five small ones?

[Pitman p. 236, #17, Modified by Mattingly]

Mixing Poisson Random Variables 1

Assume that  \(X\), \(Y\), and \(Z\) are independent Poisson random variables, each with mean 1. Find

  1. \(\mathbf{P}(X+Y = 4) \)
  2. \(\mathbf{E}[(X+Y)^2]\)
  3. \(\mathbf{P}(X+Y + Z= 4) \)

Random Errors in a Book

A book has 200 pages. The number of mistakes on each page is a Poisson random variable with mean 0.01, and is independent of the number of mistakes on all other pages.

  1. What is the expected number of pages with no mistakes ? What is the variance of the number of pages with no mistakes ?
  2. A person proofreading the book finds a given mistake with probability 0.9 . What is the expected number of pages where this person will find a mistake ?
  3. What, approximately, is the probability that the book has two or more pages with mistakes ?

 

[Pitman p235, #15]

Games with Black and White Balls

Consider the following gambling game for two players, Black and White. Black puts \(b\) black balls and White puts \(w\) white balls in a box. Black and White take turns at drawing randomly from the box, with replacement between draws until either Black wins by drawing a black ball or White wins by drawing a white ball. Suppose Black gets to draw first.

  1. Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
  2. What value of \(p\) makes the game fair (equal chances of wining) ?
  3. Is the game ever fair ?
  4. What is the least total number of balls in the game, \((b+w)\), such that neither player has more that that \(51\%\) chance of winning ?

 

[Pitman P219, #13]

Expectation of geometric distribution

Compute the expectation of the geometric distribution using the fact that in this case

\(\mathbf{E}(X)= \sum_{k=1}^{\infty} \mathbf{Pr}(X\geq k) \)

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