Solutions - 5110-02

(1) Identify distribution:

We have that $X\sim \text{Bin}(5,0.5)$, since there are $5$ trials and $p=0.5$.

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(2) Use binomial distribution formula to get PMF:

$$P_X(k)=\{\begin{array}{cc}\left(\genfrac{}{}{0px}{}{5}{k}\right){(0.5)}^k{(0.5)}^{5-k}& k\in \{0,\dots ,5\}\\ 0& \text{otherwise}\end{array}$$

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(3) Find CDF:

Since $F(x)=P[X\le x]$, we have

$$F(x)=\sum _{k=0}^{\lfloor x\rfloor }\left(\genfrac{}{}{0px}{}{5}{k}\right){(0.5)}^k{(0.5)}^{5-k}$$

Here $\lfloor x\rfloor$ or “floor” means the integer given by cutting off digits after the decimal.

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(4) Write out explicit values of CDF:

$$F_X(x)=\{\begin{array}{cc}0& x<0\\ \frac{1}{32}& 0\le x<1\\ \frac{3}{16}& 1\le x<2\\ \frac{1}{2}& 2\le x<3\\ \frac{13}{16}& 3\le x<4\\ \frac{31}{32}& 4\le x<5\\ 1& 5\le x\end{array}$$