chapter 2 - The turbulance of the forbidden city
(Geometric Distribution )
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Assuming that the emperor needs to summon a concubine to accompany him every night, after the performance of the casting, the probability of Zhen Huan being summoned is 0.3
Question 1:
Find the probability that Zhen Huan will be summoned for the first time on the 3rd day of her entry into the palace.
Question 2:
Find the probability that 3 concubines have already been summoned before Zhen Huan was summoned after entering the palace.
Geometric distribution
In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions:
The probability distribution of the number 𝑋 of Bernoulli trials needed to get one success.
The probability distribution of the number 𝑌 = 𝑋 − 1 of failures before the first success.
The geometric distribution is an appropriate model if the following assumptions are true.
The phenomenon being modelled is a sequence of independent trials.
There are only two possible outcomes for each trial, often designated success or failure.
The probability of success, p, is the same for every trial.
If these conditions are true, then the geometric random variable Y is the count of the number of failures before the first success. The possible number of failures before the first success is 0, 1, 2, 3, and so on.
An alternative formulation is that the geometric random variable X is the total number of trials up to and including the first success, and the number of failures is X − 1.
Instructions
In the Shiny app, you can learn the following knowledge:
Observe the PMF of two different geometric distributions.
Distinguish the similarities and differences between the two geometric distributions.
In the first graph, you can input the Probability of success p and the Number of failures before the 1st success x to change the diagram of the Geometric Distribution.
At the same time, users can choose the interval of the diagram, observing the Probability of the Lower tail P (X ≤ x), Upper tail P (X > x), and the Interval P (a ≤ X ≤ b).
In the second chart, you can change the Probability of success (p) and the Number of failures before the 1st success x to change the diagram of the Geometric Distribution.
Similarly, users can choose the different interval of the diagram.
There are 2 plots in the Shiny app:
1. The histogram of Geometric Distribution.
2. Red parts represent selected intervals; the rest is blue.
The following is the link to the Shiny app. Use it complete Question 1 below:
Answer
Answer
Question 1 : 0.147
Question 2 : 0.103
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