问题

Definition. Given a binomial experiment consisting of $n$ trials, the binomial random variable(二项随机变量) $X$ associated with this experiment is defined as

\[
X=\text{the number of S's among the}\ n\ \text{trials}
\]

Here we use S to denote the outcome H(heads) and F to denote the outcome T(tails).

We write $X\sim\mathrm{Bin}(n,p)$ to indicate that $X$ is a binomial rv based on $n$ trials with success probability $p$.

We denote the pmf of a binomial rv $X$ by $b(x;n,p)$. The cdf is denoted by
\[
P(X\leqslant x)=B(x;n,p)=\sum_{y=0}^{x}b(y;n,p)\quad x=0,1,2,\ldots,n.
\]

Here

\[
b(x;n,p)=\begin{cases}
\binom{n}{x}p^x(1-p)^{n-x},&x=0,1,2,\ldots,n\\
0,&\text{otherwise}.
\end{cases}
\]

Calculate the following probabilities:

• (a) $B(4; 10, .3)$
• (b) $b(4; 10, .3)$
• (c) $b(6; 10, .7)$
• (d) $P(2\leqslant X\leqslant 4)$ when $X\sim\mathrm{Bin}(10,.3)$
• (e) $P(2\leqslant X)$ when $X\sim\mathrm{Bin}(10,.3)$
• (f) $P(X\leqslant 1)$ when $X\sim\mathrm{Bin}(10,.7)$
• (g) $P(2 < X < 6)$ when $X\sim\mathrm{Bin}(10,.3)$