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Theoretical Assignment DeepBayes Summer School

2019 (deepbayes.ru)

Koreko E.

April 8, 2019

Problem 1

The random variable ξ has Poisson distribution with the parameter λ. If ξ = k we perform k

Bernoulli trials with the probability of success p. Let us define the random variable η as the

number of successful outcomes of Bernoulli trials. Prove that η has Poisson distribution with

the parameter pλ.

Solution

Use definition

where bi ∼ Ber(p) and ξ ∼ Poi(λ).

Then

As everything is (assumed) independent, conditioned on ξ = n, η ∼ B(n,p) =⇒

1

ξ is Poisson-distributed,

It follows that η is Poisson-distributed with parameter pλ.

Problem 2

A strict reviewer needs t1 minutes to check assigned application to DeepBayes summer

school, where t1 has normal distribution with parameters µ1 = 30, σ1 = 10. While a kind

reviewer needs t2 minutes to check an application, where t2 has normal distribution with

parameters µ2 = 20, σ2 = 5. For each application the reviewer is randomly selected with 0.5

probability. Given that the time of review t = 10, calculate the conditional probability that

the application was checked by a kind reviewer.

Solution

2

It is given that p(r=strict) = p(r=kind) = 0.5, p(t|r = strict) ∼ N(30,10) and p(t|r = kind) ∼

N(20,5).

When calculating normal pdf densities, in both cases

It follows that the probability the application was reviewed by kind reviewer is .

3

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