Answered

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

Awhat's P(X=2)

Bwhat's P(2\(\leq\) X \(\leq\) )

Cwhat's the expected value of X

Dwhat's the standard deviation of X

Verified Answer :

LammettHash

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

\(\mathbb P(X=2)=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac34\right)^{5-2}=\dfrac{135}{512}\approx0.26\)

\(\mathbb P(2\le X\le4)=\displaystyle\sum_{x=2}^4\mathbb P(X=x)\)
\(=\dbinom52\left(\dfrac14\right)^2\left(1-\dfrac14\right)^{5-2}+\dbinom52\left(\dfrac14\right)^3\left(1-\dfrac14\right)^{5-3}+\dbinom52\left(\dfrac14\right)^4\left(1-\dfrac14\right)^{5-4}\)
\(=\dfrac{375}{1024}\approx0.37\)

The expected value of \(X\sim\mathrm{Bin}(n,p)\) is simply \(np\), while the standard deviation is \(\sqrt{np(1-p)}\). In this case, they are \(\dfrac54=1.25\) and \(\sqrt{\dfrac{15}{16}}\approx0.97\), respectively.

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