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For X a binomial random variable with n=5 and p=1/4, answer the following

A) what's P(X=2)

B) what's P(2
\leq X
\leq )

C) what's the expected value of X

D) what's the standard deviation of X

1 Answer

4 votes
Recall that for
X\sim\mathrm{Bin}(n,p), i.e. a random variable
X following a binomial distribution over
n trials and with probability parameter
p,


\mathbb P(X=x)=f_X(x)=\begin{cases}\dbinom nx p^x(1-p)^(n-x)&\text{for }x\in\{0,1,\ldots,n\}\\\\0&\text{otherwise}\end{cases}

So you have


\mathbb P(X=2)=\dbinom52\left(\frac14\right)^2\left(1-\frac34\right)^(5-2)=(135)/(512)\approx0.26


\mathbb P(2\le X\le4)=\displaystyle\sum_(x=2)^4\mathbb P(X=x)

=\dbinom52\left(\frac14\right)^2\left(1-\frac14\right)^(5-2)+\dbinom52\left(\frac14\right)^3\left(1-\frac14\right)^(5-3)+\dbinom52\left(\frac14\right)^4\left(1-\frac14\right)^(5-4)

=(375)/(1024)\approx0.37

The expected value of
X\sim\mathrm{Bin}(n,p) is simply
np, while the standard deviation is
√(np(1-p)). In this case, they are
\frac54=1.25 and
\sqrt{(15)/(16)}\approx0.97, respectively.
User Abyshukla
by
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