MHB Binomial distribution regarding: (≤, >, etc.)

iamlorde
Messages
6
Reaction score
0
Question is as follows:

Let X be a binomial random variable with p = 0 2 .
and n = 20. Use the binomial table in Appendix A to determine
the following probabilities.
(a) P(X ≤ 3) (b) P(X > 10)
(c) P(X = 6) (d) P(6 ≤ X ≤ 11)

NK7cCqq.gif


(a) = 0.4114 is the answer. Yet all I see from this answer is that X is simple equal to "0.4114". If it is "X ≤ 3" shouldn't "0.2061", "0.0692", and "0.0115" contribute to the answer somehow because they are "<" smaller than 3?

I feel like I may be missing a fundamental element here. How do I proceed with these in general? My logic seems flawed on this matter.

For example: (c) = 0.9133-0.8042=0.1091; how is this possible. Why isn't this straight out "0.9133", the value directly next to #6?

Please enlighten me on this matter. Thank you.
 
Mathematics news on Phys.org
The numbers in the column of your table are cumulative, and so for a) you would simply read from the table to get:

a) $$P(x\le3)=0.4114$$

And for c), you would do the following:

c) $$P(x=6)=P(x\le6)-P(x\le5)=0.9133-0.8042=0.1091$$

How do you now suppose you would do parts b) and d)?
 
So, I would think,

(b) P(X > 10) = [!not](everything up to and including 10) = 1-0.9994 = 0.0006

(d) P(6 ≤ X ≤ 11) :

X is between 6 and 11, AND it includes them both, so I choose one above 11, so 12 which is: 1.0000. Then I choose one below 6, so 5, so that I can include 6. Hence: 0.8042.

Now I subtract: 1 - 0.8042 = 0.1958 (YET THIS IS FALSE)
My solution says: 0.1957?
 
Your solution for (d) is almost correct. The problem is that you've calculated $\mathbb{P}(6 \leq X \leq 12)$ because you've also included $12$. We have

$$\mathbb{P}(6 \leq X \leq 11) = \mathbb{P}(X \in \{6,7,8,9,10,11\}) = \mathbb{P}(X \leq 11) - \mathbb{P}(X \leq 5) = 0.1957$$.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top