Question

2. Let X be a binomial variable with n=10. Suppose E(X) - 3. (a) (5 points) What is the probability of success of the binomial experiment that generates the variable X? Explain your answer. (b) (5 points) Find P(X = 3). (c) (5 points) Find P(3 < X < 7). (d) (5 points) Find P(X 1 )

Answer 1

Let X be the binomial random variable with n=10 and E(X)=3

(a)

We know:

E(X) = n*p=3

+ E(X) = 10 * p = 3

+p=3/10 = 0.3

Hence the required probability is 0.3

Also

X ~ Binomial(10,0.3)

P(X = 1) = (29) + 0.3* +0,710-4

a)

P(X = 3) = (19) *0.3° +0.710-3 = 0.2668279

b)

P(3< X < 7) = P(X = 4) + P(X = 5) + P(X = 6)

-0.2001209 +0.1029193 +0.0367560 = 0.3397972

c)

P(X > 1) = 1 - P(X = 0) = 1 - *0.30 * 0.710-0

-1-0.0282475=0.9717525

Calculations:

X	P(X=x)
0	0.0282475
1	0.1210608
2	0.2334744
3	0.2668279
4	0.2001209
5	0.1029193
6	0.0367569
7	0.0090017
8	0.0014467
9	0.0001378
10	0.0000059

Binomial Graph 0.3000000 0.2500000 2 0.2000000 0.1500000 Probability P(X=x) 0.1000000 0.0500000 0.0000000 0 1 2 3 4 5 6 7 8 9 10

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