Thinking of two combinations of Bernoulli experiment

thinking of two combinations of Bernoulli experiment

The definition of the @ (probability theory) Bernoulli Test (Bernoulli experiment)

Start thinking from the most basic definition:
Bernoulli Test (Bernoulli experiment): a randomized trial performed repeatedly and independently under the same conditions . It is characterized by the possibility that there are only two possible outcomes of this randomized trial: to occur or not to occur. Then we assume that the experiment is repeated n times independently , then we call this series of repetitive independent randomized trials as N-Heavy Bernoulli tests , or Bernoulli-like.

Points
1. "Under the same conditions" is intended to indicate that the results of each trial are not affected by other experimental results. events are independent of each other .
2. The key to judging whether a test is a Bernoulli test is: first of all, it must be repeated trials, that is, multiple trials, not one test, and secondly, the results of each trial are unrelated to the results of other trials, that is, the probability of events occurring does not affect each other.

If you simply follow the definition of the question, then it is high school difficulty. That is, simply remember: x∼b (n,p), p is the probability of occurrence. X\sim B (n,p), p is the probability of occurrence. X just focuses on the occurrence or non-occurrence of a thing.

At the university level, it is necessary to be able to identify the combination of events and extract multiple Bernoulli types.

Assuming that x, Y are both Bernoulli-type, or N-times, the probability of each occurrence is p. In each variable to do n times, can do two together, so that only n times, it implies two Bernoulli approximate type. Yes, only X, Y is incompatible.

Let's look at an exercise.

(2016-8) Random trial E has three kinds of 22 incompatible results a1,a2,a3 a_1, a_2, a_3, and three kinds of results occur in the probability of \frac{1}{3}, the test E is repeated 2 times, X indicates the results of 2 tests A1 a_1 hair Number of times, y indicates the number of times the result A2 a_2 occurred in 2 trials, the phase of X and Y
The coefficient of −12⎯⎯⎯⎯⎯⎯⎯\underline{-\frac{1}{2}}

Analysis: Randomized trials with three different 22 incompatible results. We stand on every result to see the problem. The probability of each result occurring is \frac{1}{3}, the probability of not occurring is \frac{2}{3}
So the number of times the result of the N-Test is the Bernoulli generalization. now there are three results, and they will not occur simultaneously , that is, incompatible, so this is the combination of three Bernoulli types under a N-weight test.

Knowing this, the problem will be very simple.

X∼b (2,13) →EX=NP=23,DX=NP (1−p) =49 X\sim B (2,\frac{1}{3}) \rightarrow EX = NP = \FRAC{2}{3},DX = NP (1-P) = \frac{4}{9}

Y∼b (2,13) →ey=np=2<