Question

Let x be a binomial random variable with n=20 and p = 0.1.

a. Find the formula for the probability distribution of x.

b. Calculate P(X≤4)

c. Calculate the mean and standard deviation of X.

Answer 1

$n=20, \ p=0.1,\ q=1-p=1-0.1=0.9.$

a. The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:

$P(X=k)=\begin{pmatrix} n \\ k \end{pmatrix}\cdot p^k \cdot q^{n-k},$where $\begin{pmatrix} n \\ k \end{pmatrix}=\cfrac{n! } {k! \cdot(n-k)! }$

is the binomial coefficient.

$P(X=k)=\\=\begin{pmatrix} 20 \\ k \end{pmatrix}\cdot 0.1^k \cdot 0.9^{n-k} =\\ =\cfrac{20! } {k! \cdot(20-k)! } \cdot 0.1^k \cdot 0.9^{n-k} .$

b.

$P(X\le4) =\\P(X=0)+P(X=1)+\\+P(X=2)+P(X=3)+\\+P(X=4)=\\ =\cfrac{20!}{0!\cdot20!}\cdot0.1^0\cdot0.9^{20} +\\+ \cfrac{20!}{1!\cdot19!}\cdot0.1^1\cdot0.9^{19} +\\+ \cfrac{20!}{2!\cdot18!}\cdot0.1^2\cdot0.9^{18} +\\+ \cfrac{20!}{3!\cdot17!}\cdot0.1^3\cdot0.9^{17} +\\+ \cfrac{20!}{4!\cdot16!}\cdot0.1^4\cdot0.9^{16} =\\ =0.9568.$

c. The mean

$$\mu=np=20\cdot0.1=2.$$

The standard deviation

$$\sigma=\sqrt{npq} =\sqrt{20\cdot0.1\cdot0.9} =1.34.$$