Verify and Explain Binomial R.V. Identities

This will show that the terms for P(Y=n-k) to P(Y=n) are exactly the same as the terms for P(X=k) to P(X=0). Therefore, the sums are equal, which means that P(Y≥n-k) = P(X≤k). In summary, the identities for binomial random variables (n,p) and (n,1-p) are verified and explained as follows: a.) P{X<=i}= P{Y>=n-i}, which can be proven by examining the terms of the binomial distribution function and noticing that they are the same in opposite order. b.) P{X=k}= P{Y=n-k}, which can be proven by replacing
  • #1
knowLittle
312
3
If X and Y are binomial random variables with respective parameters (n,p) and (n,1-p), verify and explain the following identities:
a.) P{X<=i}= P{Y>=n-i};
b.) P{X=k}= P{Y=n-k}

Relevant Equations:
P{X=i}=nCi *p^(i) *(1-p)^(n-i), where nCi is the combination of "i" picks given "n".

Distribution function
##p\left\{ x\leq i\right\} =\sum _{k=0}^{i}\left( n_{k}\right) p^{k}\left( 1-p\right) ^{n-k}
##, where n_k is ## (_{k}^{n})
##
Solution:
I don't know, how to start the problem. Please, help.
 
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  • #2
If you examine the terms of the binomial you will see that the terms of (n,p) and (n,1-p) are the same except they are in opposite order from each other..
 
  • #3
What do you mean by opposite order?
 
  • #4
I know that * p and 1-p * are complement of each other, but how does this help me?
 
  • #5
knowLittle said:
What do you mean by opposite order?

Term of index k for (n,p) (a) = term of index n-k for (n,1-p) (b). As a result the problems you are addressing involve terms from (n,p) being equal to the corresponding (opposite order) terms in (n,1-p).

(a) {n!/[k!(n-k)!]}pk(1-p)n-k

(b) {n!/[(n-k)!k!]}(1-p)n-kpk

Notice that (a) and (b) are exactly the same.
 
  • #6
From what I posted originally:
If X and Y are binomial random variables with respective parameters (n,p) and (n,1-p), verify and explain the following identities:
a.) P{X<=i}= P{Y>=n-i};
b.) P{X=k}= P{Y=n-k}

I have proven part b.) and yes it's the same.
How do I prove part a.)? Particularly, P{Y>=n-i}.

I know that
P{Y>=n-i}=1-P{Y=0}-... up until Y=**A number lesser than (n-i)-1**

Please, help.
 
  • #7
For part a, the X probability is gotten by adding all the terms from the beginning to k, while the Y probability is gotten by adding all the terms from n-k to the end. Since these terms match term by term, the sums match.
 
  • #8
mathman said:
...the Y probability is gotten by adding all the terms from n-k to the end. Since these terms match term by term, the sums match.

What do you mean by " the end"?
I know that P{Y>=n-k} has to be described as
1-P{Y=0}-...-P{Y=n-k-1}

Could you develop further on what you said?
 
  • #9
P(Y≥n-k) = P(Y=n-k) + P(Y=n-k+1) + ... + P(Y=n).

P(Y=n-k) = P(X=k), P(Y=n-k+1) = P(X=k-1), ..., P(Y=n) = P(X=0).

Therefore P(Y≥n-k)=P(X≤k).
 
Last edited:
  • #10
I understand:
mathman said:
P(Y≥n-k) = P(Y=n-k) + P(Y=n-k+1) + ... + P(Y=n).

But, I do not understand this part:
mathman said:
P(Y=n-k) = P(X=k), P(Y=n-k+1) = P(X=k-1), ..., P(Y=n) = P(X=0).
Therefore P(Y≥n-k)=P(X≤k).

I know by one of the identities that P(Y=n-k) = P(X=k), but I don't know from where do you get the rest.
 
  • #11
knowLittle said:
I understand:I know by one of the identities that P(Y=n-k) = P(X=k), but I don't know from where do you get the rest.

Use the property of the combination. In other words what are the properties of combinations in pascal's triangle? (What is nCk vs nC[n-k] and similar properties in relation to your question)?
 
  • #12
knowLittle said:
I understand:


But, I do not understand this part:


I know by one of the identities that P(Y=n-k) = P(X=k), but I don't know from where do you get the rest.
Replace k, term by term, by k-1, k-2, etc. until you get to 0.
 

What is a binomial random variable?

A binomial random variable is a discrete random variable that represents the number of successes in a series of independent trials, where each trial has only two possible outcomes: success or failure. It follows a binomial distribution, which can be used to calculate the probability of a specific number of successes in a given number of trials.

What are the properties of a binomial random variable?

There are three main properties of a binomial random variable: 1) there are a fixed number of trials, 2) each trial has only two possible outcomes, and 3) the trials are independent of each other. These properties allow us to use binomial distributions to model real-world situations with binary outcomes.

How do you verify a binomial random variable identity?

To verify a binomial random variable identity, we can use the binomial theorem, which states that the sum of the probabilities of all possible outcomes in a binomial distribution is equal to 1. We can also verify the identity by using the formula for calculating the probability of a specific number of successes in a given number of trials.

What is the difference between a binomial random variable and a binomial distribution?

A binomial random variable is a discrete random variable that represents the number of successes in a series of independent trials, while a binomial distribution is a probability distribution that describes the likelihood of obtaining a specific number of successes in a given number of trials. In other words, the random variable is a concept, while the distribution is a mathematical function.

How do you explain the binomial random variable identities?

The binomial random variable identities are mathematical expressions that represent the probabilities of obtaining a specific number of successes in a given number of trials. These identities can be derived from the binomial distribution formula and are useful for calculating the probabilities of different outcomes in binomial experiments.

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