CK-12 Probability and Statistics - Advanced

7.3. Binomial Distribution and Binomial Experiments http://www.ck12.org

k P(X=k)

0 1 / 16

1 4 / 16

2 6 / 16

3 4 / 16

4 1 / 16

The expected value for the above distribution is:

E(X) = 0 ( 1 / 16 )+ 1 ( 4 / 16 )+ 2 ( 6 / 16 )+ 3 ( 4 / 16 )+ 4 ( 1 / 16 )

E(X) =2 In other words, you expect half of the four to guess correctly when given two equally, likely choices.
E(X)can be written as 4·^12 which is equivalent ton p.For a random variableXhaving a binomial distribution with
ntrials and probability of successesp, the expected value (mean) and standard deviation for the distribution can be
determined by:

E(X) =n p=μx and σx=

√

n p( 1 −p)

The graphing calculator will now be used to graph and compare binomial distributions. The binomial distribution
will be entered into two lists and then displayed as a histogram. First we will use the calculator to generate a
sequence of integers and secondly the list of binomial probabilities.

Sequence of integers:

2 nd[LIST]→OPS↓ 5 .seq ( ) STO→ 2 nd 1

Binomial Probabilities:

2 ndDISTR 0:binompdf(

Horizontally, the following are examples of binomial distributions wherenincreases and p remains constant.
Vertically, the examples display the results wherenremains fixed andpincreases.

n=5 and p= 0. 1 n=10 and p= 0. 1 n=20 and p= 0. 1

For the small value ofp, the binomial distributions are skewed toward the higher values ofx. Asnincreases, the
skewness decreases and the distributions gradually move toward being more normal.

n=5 and p= 0. 5 n=10 and p= 0. 5 n=20 and p= 0. 5