Home » Posts tagged 'JCM_math230_HW6_S13'
Tag Archives: JCM_math230_HW6_S13
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.
- The diamond with vertices at \( (0,2), (-2,0), (0,-2), (2,0) \).
- 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 \(c>0\).
- Find \( c \).
- Find \(\mathbf{E}(X)\).
- Find \(\mathrm{Var}(X) \).
Raindrops are falling
Raindrops are falling at an average rate of 30 drops per square inch per minute.
- What is the chance that a particular square inch is not hit by any drops during a given 10-second period ?
- 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?
- 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
- \(\mathbf{P}(X+Y = 4) \)
- \(\mathbf{E}[(X+Y)^2]\)
- \(\mathbf{P}(X+Y + Z= 4) \)
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.
- What is the expected number of pages with no mistakes ? What is the variance of the number of pages with no mistakes ?
- 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 ?
- 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.
- Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
- What value of \(p\) makes the game fair (equal chances of wining) ?
- Is the game ever fair ?
- 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
\(\mathbf{E}(X)= \sum_{k=1}^{\infty} \mathbf{Pr}(X\geq k) \)