NOTE: Please complete Number 4 by Thursday! (The rest of the homework is due Tuesday, April 13, at 11:59 pm).


  1. Suppose that the time until the next earthquake in a particular place in California has an exponential distribution with expectation 2 years. Find the probability that the next earthquake happens within
  1. one year.
  2. six months.
  3. ten years.



2. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter \(\lambda = 1\). What is

  1. the probability that the repair time exceeds 2 hours?
  2. The probability that the reapir takes at least 10 hours, given that its duration exceeds 9 hours?


3. Suppose you have a very long highway which we will model as a straight line of infinite length starting at \(x=0\) and stretching all the way to \(x=\infty\). One fire station it to be located along the road and a fire will break out at a random point along the road. The distance \(X\) of the fire from \(X=0\) has an exponential distribution with mean equal to 4 miles.

  1. Write down the pdf of the random variable \(X\).
  2. What is the variance of \(X\). You can use the formula derived in class, no need to re-derive it.
  3. We want to find the optimal place \(x=a\) to locate a fire station on this road. To do this we wish to minimize the funtion \[H(a) = E[(X-a)^2], \ \ \ a\geq 0.\] Choose \(a\) to minimize this function.


4. We are starting the part of the course where we require multivariable calculus. In particular we need to be comfortable doing double integrals of various types. I have put links on ‘Sakai/Resources/Lecture Slides/Multivariable Calculus Review’: that are Kahn Academy refreshers on multivariable functions, graphs, and integrals. Please review these to the extent you need a review of MATH 233 You will receive credit for this as follows:

  1. Have you reviewed/skimmed the link 1. Khan Academy - Multivariable functions? (Yes = 5 pts, NO = 0 pts)
  2. Have you reviewed/skimmed the link 2. Khan Academy - Multivariable Graphs? (Yes = 5 pts, NO = 0 pts)
  3. Have you reviewed/skimmed the link 3. Khan Academy - Double Integrals? (Yes = 10 pts, NO = 0 pts)


  1. Compute \(\int_{y=0}^1 \int_{x=0}^2 \cos{(x+y)}dxdy\).


  1. Consider the region \[A = \left\{(x,y):0<x<1, 0<y<2, x^2 <y \right\}\] You will compute \[\int \int_A (x^2+xy) dxdy\] in two ways.
  1. First graph the region \(A\) in the \(xy\)-plane.
  2. Integrate with respect to \(x\) first and then with respect to \(y\).
  3. Integrate with respect to \(y\) first and then with respect to \(x\).


  1. Suppose the random variables \((X,Y)\) have joint pdf given by \[f(x,y) = \left\{\begin{array}{ll} cx^2y^2 & \text{if } 0\leq x\leq y \leq 1,\\ 0 & \text{otherwise.}\\ \end{array} \right. \]
  1. Find the constant \(c\).
  2. Find the marginal density of \(X\).
  3. Find the marginal density of \(Y\).
  4. Are \(X\) and \(Y\) independent?


  1. Suppose the random variables \((X,Y)\) have joint pdf given by

\[f(x,y) = \left\{\begin{array}{ll} x+y & \text{if } 0<x<1 \text{ and } 0< y < 1,\\ 0 & \text{otherwise.}\\ \end{array} \right. \]

  1. Find the marginal density of \(X\).
  2. Find the marginal density of \(Y\).
  3. Are \(X\) and \(Y\) independent?
  1. Suppose \((X,Y)\) have joint pdf given by

\[f(x,y) = \left\{\begin{array}{ll} c(x^2+y^2) & \text{if } 0\leq x \leq y \leq 1,\\ 0 & \text{otherwise.}\\ \end{array} \right. \]

  1. Find the constant \(c\).
  2. Find the marginal density of \(X\).
  3. Find the marginal density of \(Y\).
  4. Find \(E[X]\)
  1. Shankar is giving a one hour talk on his research at a big university. If \(X\) denotes the number of questions he is asked by audience members in his talk then suppose \(X\) has probability mass function \[p_X(0) = \frac{1}{10},\ \ \ \ \ \ \ \ p_x(1) = \frac{4}{10},\ \ \ \ \ \ \ \ p_x(2) = \frac{4}{10},\ \ \ \ \ \ \ \ p_x(3) = \frac{1}{10}\]

Every question he gets, Shankar is able to adequately answer with probability 4/5, independent across questions. Let \(Y\) denote the total number of “adequately” answered questions.

  1. Find the joint probability mass function of the random variables \((X,Y)\) namely fill in the following table: Here the \((x,y)\) cell represents \(P(X=x, Y=y)\). Thus for example, the shaded cell is \(p_{X,Y}(2,0) = P(X=2, Y=0)\). You need to find all such probabilities and fill in the table above. Round to two decimal places.
  2. Find the marginal distribution (pmf) of \(X\) (so a table with the values X takes and the probability of taking these values). Round to two decimal places.
  3. Find marginal distribution (pmf) of \(Y\) (so a table with the values X takes and the probability of taking these values). Round to two decimal places.
  4. Find \(E[Y]\). Round to two decimal places.


  1. In many organisms (including humans), genes occur in pairs. Consider a gene of interest (for example, the gene that codes for Hemoglobin in the blood). Call the two types of this gene (referred to as alleles) as type \(a\) and Type \(A\). For each individual, the genotype of that individual is the types of the two genes in the pair. Thus there are three types of individuals: \(AA\), \(Aa\), and \(aa\) (\(aA\) is equivalent to \(Aa\)). By the Hardy-Weinberg principle (see https://en.wikipedia.org/wiki/HardyWeinberg_principle), the probability an individual of each type is gien by the table Here, \(p\) is a fixed number with \(0 < p < 1\). Now suppose you take a random sample of \(150\) people. Let \(X_1,X_2\) and \(X_3\) denote the number of individuals in the sample with genotypes \(AA\), \(Aa\) and \(aa\), respectively.
  1. Write down the joint pmf of \((X_1,X_2,X_3)\). The pmf will also involve the parameter \(p\) above.
  2. Write down the pmf of the number of people in the sample who have at least one \(A\) (i.e pmf of the random variable \(X_1 + X_2\)). Note: if you understand the premise of the multinomial you do not really need to do any calculations. The pmf will also involve the parameter p above.


12. Let \(X\) and \(Y\) be two continuous random variables with joint density given by \[f(x,y) = \left\{\begin{array}{ll} \frac{1}{4} & \text{if } -1\leq x \leq 1 \text{ and } -1\leq y \leq 1,\\ 0 & \text{otherwise.}\\ \end{array} \right. \] Compute the following probabilities:

  1. \(P(X+Y \leq \frac{1}{2})\),
  2. \(P(X-Y \leq \frac{1}{2})\),
  3. \(P(XY \leq \frac{1}{2})\),
  4. \(P(\frac{Y}{X} \leq \frac{1}{2})\).