Math Insight

Review problems for exam 3

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  1. The number of objects $N(t)$ follows a birth-death process with the following dynamics. Each object has a probability per second $0.009$ of duplicating and a probability per second of $0.01$ of disappearing. We start with $N(0)=5$ objects.
    1. If we approximated the dynamics of $N(t)$ as deterministic, what equation would describe the dynamics of $N$? What is the solution of the equation. As $t$ increases to a large number, what would happen to the number of objects $N(t)$?
    2. What is the stochastic equation for the dynamics of $N(t)$?
    3. What is the probability per second that any event occurs?
    4. What is the probability that the next event is a duplication of an object? What is the probability that the next event is a disappearance of an object?
    5. Let $T$ be the time between two consecutive events. What type of random variable is $T$?

      Imagine that at some time $t$, there are $N(t) = 4$ objects. At this moment, what is the expected value (or average) of $T$? At this moment, what is more likely: that $T$ is between $0$ seconds and $10$ seconds or that $T$ is between $10$ seconds and $20$ seconds?

  2. The curcurbit beetle (Diabrotica speciosa) show reduced mating behavior in the presence of falling air pressure, so their behavior could be used to predict weather like rain. Imagine that it rains 10% of the days. Imagine that on days when it does eventually rain, you observe reduced mating activity 90% of the time. On days when it never rains, you observe reduced mating activity only 15% of the time. You now observe reduced mating activity in the beetles. What is the probability it will rain today?

  3. Each evening, Hunter watches one television show before bed. On 30% of the evenings, he watches a drama show. On 10% of the evenings, he watches a romance show. On 60% of the evenings, he watches a horror show. On nights after watching a drama show, Hunter has a 60% chance of having a nightmare. On nights after watching a romance show, he has a 90% chance of having a nightmare. On nights after watching a horror show, he has a 70% chance of having a nightmare.
    1. Write a contingency table showing the probabilities for the different combinations of shows and nightmares. Be sure to total each row and each column.
    2. What is the probability that Hunter has a nightmare any given night?

    3. Last night, Hunter had a nightmare. What is the probability that he watched a drama show that evening? A romance show? A horror show?

  4. Let $F$ be the number of applications of a given fungicide in a field in one year. Let $D$ be the event of significant crop damage that year due to the fungus. Let $R$ be the event that the fungus develops resistance to the pesticide.
    1. Explain in words what each of these probabilities mean.
      1. $P(F=2)$
      2. $P(F=3, D)$
      3. $P(F \le 4, D^c)$
      4. $P(R \,|\, F=4)$
      5. $P(F \gt 3 \,|\, R)$
    2. In words, what do $P(F \lt 3, D)$, $P(F \lt 3 \,|\, D)$ ,and $P(D \,|\, F \lt 3)$ mean?

      Calculate a formula for $P(F \lt 3 \,|\, D)$ in terms of $P(D \,|\, F \lt 3)$.

      Calculate a formula for $P(D \,|\, F \lt 3)$ in terms of $P(F \lt 3, D)$.

    3. Imagine that 5% of the years, you apply the fungicide once, 75% of the years, you apply the fungicide twice, 14% of the years, you apply the fungicide three times, and 6% of the years, you apply the fungicide four times.

      Sketch a the probability distribution of the random number $F$.

  5. An invasive plant has begun to invade a patch of wildflowers. Assume that the each plant divides into two with probability per hour of 0.003 and that the probability per hour that it dies is 0.001.
    1. Write down a continuous time stochastic model for the dynamics of $N(t)$.
    2. Write down a deterministic model that approximates the dynamics of $N(t)$.
    3. Solve the deterministic model assuming that when $t=0$, there are 7 plants.
    4. How close to do expect the deterministic model to be to the stochastic model? What important features of the behavior of $N(t)$ are missing in the deterministic model?
    5. Solve the deterministic model assuming that when $t=0$, there are 950 plants. In this case, how close do you expect the deterministic model to approximate the stochastic model. Justify your answer.
    6. Assume that the patch of land could hold at most $K=1170$ plants. Assume that this carrying capacity $K$ modifies both the birth rate and the death rate, multiplying them by the factor $\left(1-\frac{N(t)}{K}\right)$.

      Modify both the stochastic model and the deterministic model to include the effect of a carrying capacity of $K=1170$ plants.