Course Landing Page

W05 - HW Solutions 1

4 min read

Poisson process

05-01 - Poisson satisfies

Show that a Poisson variable satisfies the total probability rule for a CDF, namely that .

Solution

  1. & State CDF of a Poisson distribution.
    • We know that
    • We know that .
  2. & Compute limit as .
    • Note that

05-02 - Expectation of Poisson

Derive the formula for a Poisson variable .

Solution

  1. & State PMF of a Poisson distribution.
  2. & Find expectation.
    • We know that .

05-03 - Application of Poisson: meteor shower

The UVA astronomy club is watching a meteor shower. Meteors appear at an average rate of 4 per hour.

  • (a) Write a short explanation to justify the use of a Poisson distribution to model the appearance of meteors Why should appearances be Poisson distributed?
  • (b) What is the probability that the club sees more than 2 meteors in a single hour?
  • (c) Suppose that over a four hour evening, 13 meteors were spotted. What is the probability that none of them happened in the first hour?

Solution (a)

  1. & Write explanation.
    • You use Poisson distribution if events occur randomly and you know the mean number of events that occur within a given interval of time.
    • In addition, Poisson distributions are advantageous when describing rare events. Since meteors are a rare occurrence, it makes sense to use a Poisson distribution. (b)
  2. & Compute probability.
    • Since , it’s easier to compute the latter. (c)
  3. & Compute probability.
    • We know that there are 13 meteors in 4 hours, so we see an average of meteors per hour. Let
    • We wish to find the probability .

05-04 - Silver dimes

Suppose 1 our of 350 dimes in circulation is made of silver. Consider a tub of dimes worth 40$.

  • (a) Find a formula for the exact probability that this collection contains at least 2 silver dimes. Can your calculator evaluate this formula?
  • (b) Estimate the probability in question using a Poisson approximation.

Solution (a)

  1. & Identify distribution.
    • Clearly, this scenario follows a binary distribution.
    • We have a chance that the dime is made of silver.
    • Since we have 40$ worth of dimes, there are 400 dimes.
    • Thus, .
  2. & Find formula for probability.

(b)

  1. & Find corresponding Poisson distribution.
    • Let .
    • .
  2. & Compute probability.
    • We have that .

05-05 - Applications of Poisson approximation of binomial

Let and consider the Poisson approximation to .

  • (a) Estimate the possible error of the approximation (for an arbitrary probability).
  • (b) Compute the exact error of the approximation for the specific value .

Solution (a)

  1. & Define random variable that is the Poisson approximation to .
    • .
  2. & Estimate error.

(b)

  1. && Compute , the exact value.
  2. & Compute .
  3. & Compute difference.

05-06 - Constants in PDF from expectation

Suppose has PDF given by:

Misplaced &a + b x^{2} & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ Suppose $E[X] = \frac{7}{10}$. Find the only possible values for $a$ and $b$. Then, find $\text{Var}[X]$. **Solution** 1. && Integrate expression for $E[X]$. $$E[X] = \int_{0}^{1}x\left(a+bx^{2}\right) dx= \left.\left[\frac{ax^{2}}{2} + \frac{bx^{4}}{4}\right]\right|^{1}_{0} = \frac{2a + b}{4} = \frac{7}{10}$$ 2. & Integrate original PDF. $$\int_{0}^{1}a+bx^{2}dx = \left.\left[ax + \frac{bx^{3}}{3}\right]\right|_{0}^{1} = a + \frac{b}{3} = 1$$ 3. & Solve system of equations for $a$ and $b$. - We have that $a = 1 - \frac{b}{3}$. - Plugging into the first equation yields $b = \frac{12}{5}$ - Plugging $b = \frac{12}{5}$ into the second equation yields $a = \frac{1}{5}$. 4. & Find $E\left[X^{2}\right]$. $$E\left[X^{2}\right] = \dots$$