Question sets-Discrete Distribution Family

Question 1

BIOF2014 consists of 20 multiple choice questions, with 4 choices for each question. A student goes into the examination and tries o randomly guess the answers It is obvious that the total correct answers the student can get follows Binomial distribution.

  1. How likely can the student get exactly 10 correct answer?

\text{p-value}=\binom{20}{10}0.25^{10}0.75^{10}

  1. If each correct answer can get 5 marks, what is the expect grade of the student?

\text{E}(5x)=5*20*0.25=25

Question 2

During the transportation process, The probability of an egg is cracked is 0.01

  1. For a box of 8 eggs how likely can a box have at least one cracked egg?

P(X\geq 1) =1- P(x=0)= 1- \binom{8}{0}0.99^80.1^0=0.08

  1. If a product quality control is checking the quality of the eggs to ensure all eggs are not cracked, and in average how may boxes does he need to examine to find a box with at least a cracked egg? . What will be the probability distribution to model this outcome

E(x)=\frac{1}{0.08}=12.5

  1. A product quality control is to continuously monitoring the quality of the eggs. They would liek to have a short review of the case once they have observed 4 boxes of egg with cracks. in average how may boxes does he need to examine to commence the review? What will be the probability distribution to model this situation?

E(x)=\frac{4*(1-0.08)}{0.08}

Question 3

For a X\sim G(p), please show the following property

P(X>s|X>t)=P(X>s-t) \quad \forall s>t

\begin{align*} P(X>s|X>t)&=\frac{P(X>s \text{ and } X>t)}{p(X>t)} =\frac{P(X>s)}{p(X>t)}\\ &=\frac{p(1-p)^{s-1}}{p(1-p)^{t-1}} = (1-p)^{s-t}\\ &=P(X>s-t) \end{align*}

Question 4: Wright-Fisher Model

Suppose the population size is N=50 diploid individuals (so the total number of alleles is 2N=100). In the parent generation, the frequency of allele A is 0.4 (meaning n_{A0} = 40). Using the formula provided in the slides, write the expression for the probability that the number of A alleles in the next generation (n_{A1}) will remain exactly 40.

P(n_{A1}=40) = \binom{100}{40} (0.4)^{40} (0.6)^{60}

Question 5: Coalescent Theory

If the effective population size N_e = 1000, calculate the probability that two randomly selected alleles coalesce exactly 3 generations ago (t=3).

P(T=3) = (1 - 1/2000)^2 \times (1/2000)

Question 6: Clinical trial

You are analyzing a contingency table from a clinical trial comparing a Treatment vs. a Control group. Using the Hypergeometric probability mass function, calculate the probability of observing exactly a=4.

Improvement (Y) No Improvement (Not Y)
Treatment a=4 b=6
Control c=2 d=8

P(X=4) = \frac{\binom{10}{4}\binom{10}{2}}{\binom{20}{6}}

Question 7:

why the combination term in the Negative Binomial PMF is \binom{x-1}{r-1} rather than \binom{x}{r}

Because the last trial must be a success. Therefore, we only need to arrange the previous r-1 successes among the previous x-1 trials.