NOTE: This homework is due on Tuesday, March 31. Please turn in your solutions on Gradescope. There is a chance that I will add a couple of problems on Thursday.


  1. In a scientific experiment the measurement error made by an instrument is a random variable \(X\) with probability density function \[f_X(x) = \left\{ \begin{array}{ll} cx^2 & \text{if } -3\leq x\leq3,\\ 0 & \text{otherwise} \\ \end{array} \right.\]
  1. Find the value of c. Leave your answer as a fraction.
  2. Calculate \(P(|X|=1/2)\). Here \(|X|\) is the absolute value of \(X\).
  3. Calculate \(P(|X| > 2)\).



2. Random variables with densities as in the example below turn out to be very important in a number of disciplines including social networks [1,3], modeling complex systems such as wealth distribution, the sizes of cities [4] as well as in mathematical theories of evolution [5].

\[f_X(x) = \left\{ \begin{array}{ll} c/x^3 & \text{if } x \geq 10,\\ 0 & \text{otherwise} \\ \end{array} \right.\]

  1. Find the value of \(c\). Round to four decimal places.
  2. Calculate \(P(X=15)\).
  3. Calculate \(P(X > 2)\).
  4. Calculate \(P(X^3 > 200)\).


3. Luke is going to a party in a neighboring town (Durham). Suppose that if he is \(s\) minutes early for the party, then he incurs an emotional cost \(4s\) (time spent twiddling thumbs looking uncomfortably at hosts, etc.), and if he is \(s\) minutes late, then he incurs a cost \(2s\) (missing out the funnest part of the party, if one is too late cliques already form and one stands around looking uncomfortably at the groups already having tons of fun). Suppose also that the travel time \(X\) measured in minutes from where he is presently are to the location of the party is a continuous random variable having probability density function

\[f_X(x) = \left\{ \begin{array}{ll} \frac{1}{1800}(60-x), & \text{for } 0 <x < 60,\\ 0 & \text{otherwise} \\ \end{array} \right.\]

Determine the time at which Luke should depart if he wants to minimize his expected cost. You can use the expression we derived in class and go from there.


4. A big dam is constructed above a town. If the dam collapses and water flows down the hill into the town, the loss (measured in millions of dollars) is a random variable \(X\) with density function

\[f_X(x) = \left\{ \begin{array}{ll} c(10-x), & \text{for } 0 <x < 10,\\ 0 & \text{otherwise} \\ \end{array} \right.\]

  1. Find \(c\). Round to four decimal places.
  2. Suppose you are told that the loss is more than (\(>\)) 6. Find the probability the loss is at least (\(\geq\)􏰏) 8. Round to four decimal places.


5.A uniform random variable in the interval (0, 1) is defined as a random variable with density

\[f_X(x) = \left\{ \begin{array}{ll} 1, & \text{for } 0 <x < 1,\\ 0 & \text{otherwise} \\ \end{array} \right.\]

  1. Find \(E\left[e^{2x}\right]\).
  2. Find \(E\left[e^{5x}\right]\).


6. Using L’Hopital’s rule (or otherwise) find the following limits. (You may have to dredge up your recollections of Calculus I.)

  1. \[\lim_{t\rightarrow 0} \frac{e^t-1}{t}\]
  2. \[\lim_{t\rightarrow 0} \frac{e^t-1}{e^{2t}-1}\]
  3. \[\lim_{t\rightarrow 0} \frac{\ln t}{t^2-1}\]


7. Suppose X is a uniform random variable in the interval \((−2,10)\). Calculate \(E(X)\) \(Var(X)\). You can use the formulas we learned in class directly. No need to derive.


8. You arrive at a bus stop at 10:00 am knowing that the bus will arrive at some time uniformly distributed between 10:00 and 10:30.

  1. What is the probability that you will have to wait longer than 10 minutes?
  2. If, at 10:15, the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?