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I struggled to come up with an approach and I'm not sure if induction is needed. Please help!

EDIT:

Thanks for all the quick feedback!

Is there a way to prove that for a random variable x with a binomial distribution, P(x=0) + P(x=1) + ... + P(x=n) = 1? Possibly through induction?

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    $\begingroup$ It is discrete by its very definition, there is nothing to be proven. $\endgroup$ – Yves Daoust Jun 6 at 7:26
  • $\begingroup$ How can you use induction on non integer points? $\endgroup$ – Upstart Jun 6 at 7:29
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    $\begingroup$ Write down the definition of a binomial distribution. Then write down your definition of a discrete distribution. The rest should be obvious. $\endgroup$ – Theoretical Economist Jun 6 at 7:49
  • $\begingroup$ Thanks for all the quick feedback! Is there a way to prove that for a random variable x with a binomial distribution, P(x=0) + P(x=1) + ... + P(x=n) = 1? Possibly through induction? $\endgroup$ – Shukie Jun 6 at 8:01
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A random variable X is said to have the Binomial distribution with parameters $N,p$ if

$p(X =x) = \begin{cases} \binom{N}{x} \,\,p^x \,\,(1-p)^{N-x} & 0 \leq x \leq N \\\\ 0 & otherwise \end{cases} $

since the support of random variable is $\{0,1,2..,N\}$.we can conclude that it is discrete distribution

EDIT:

following is the classic (and elegant) proof if you know Binomial expansion

$ = \sum_{x=0}^N \binom{N}{x} \, p^x \, (1-p)^{N-x}$

$ =\Big( p + 1-p \Big)^N$ ( By binomial expansion )

$ = 1^{N}$

$ = 1$

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  • $\begingroup$ oops sorry.edited :) $\endgroup$ – viru Jun 6 at 7:45
  • $\begingroup$ Thanks for all the quick feedback! Is there a way to prove that for a random variable x with a binomial distribution, P(x=0) + P(x=1) + ... + P(x=n) = 1? Possibly through induction? $\endgroup$ – Shukie Jun 6 at 8:01
  • $\begingroup$ I have added you can check now. $\endgroup$ – viru Jun 6 at 16:27
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If $X$ has binomial distribution with parameters $n$ (and $p$) then $P(X\in\{0,1,\dots,n\})=1$.

The set $\{0,1,\dots,n\}$ is a countable set.

This together states that binomial distribution can be labeled as a discrete distribution.

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