𝑃 ( 𝑋 ≥ 𝑘 − 1 ) for X∼Binomial(n−1,p).
P(X≥k) for X∼Binomial(n,p)
In probability theory, cumulative distribution functions (CDFs) allow us to understand the distribution of random variables by showing the probability that a random variable is less than or equal to a certain value.
If you have two random variables, X and Y, each with their own CDF Fx(x) and Fy(y), you can compare them at a specific point x (i.e., comparing Fx(x) and Fy(x)). However, the outcome depends on the values of x and the specific distributions of X and Y.
In general, you can determine which term is greater in terms of their probability:
To summarize, whether you can say which CDF is greater depends on the values of x you're evaluating and the specific distributions involved.
Hi,
We know that if X1~Bin(n-1,p) and X2~Bin(1,p) then Y=X1+X2~Bin(n,p).
On the other hand if the events A and B be defined as A={X1=k+1), (1).
On the other hand we have
P(X1>=k)>P(X1>=k+1), (2).
Comparing (1) and (2) yields
P(X1>=k)>P(X1+X2>=k+1)
Regard,
Hamid
Good proof, Hamid Ghorbani,
the only needed clarification is the fact that X1 and X2 must be independent binomial variables.
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