Question

Let descrete random variable X - Bin(9,0.3) Find: 1) Probability P(X>5) 2) Probability P( X 2 ) 3) Probability P(2<x<5) 4) Probability P(2<x<5) 5) Probability P(X=0) 6) Probability P(X= 7) 7 Mx 8) Ox Show your explanations. Displaying only the final answer is not enough to get credit Note: round calculated numerical values to the fourth decimal place where applicable.

Answer 1

Let ,

X + Bin = 9, p=0.3)

Therefore , The PMF of X is ,

; x=0,1,2,...........,n and q=1-p

P(X = 1) =

= 0 ; otherwise

The probability distribution table is ,

X
0	1	1	0.0404	0.0404
1	9	0.3	0.0576	0.1556
2	36	0.09	0.0824	0.2668
3	84	0.027	0.1176	0.2668
4	126	0.0081	0.1681	0.1715
5	126	0.0024	0.2401	0.0735
6	84	0.0007	0.343	0.021
7	36	0.0002	0.49	0.0039
8	9	7E-05	0.7	0.0004
9	1	2E-05	1	2E-05

1)

P(X >5) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)

$=0.0210+0.0039+0.0004+0.0000=0.0253$

2)

$P(X\geq 2)=1-P(X<2)=1-[P(X=2)+P(X=1)]$

$=1-[0.1556+0.0404]=0.8040$

3)

$P(2\leq X<5)=P(X=2)+P(X=3)+P(X=4)$

$=0.2668+0.2668+0.1715=0.7051$

4)

$P(2<X\leq 5)=P(X=3)+P(X=4)+P(X=5)$

$=0.2668+0.1715+0.0735=0.5118$

5) P(X=0)= 0.0404

6) P(X=7)=0.0039

7) $\mu_{X}=np=9*0.3=2.7$

8) $\sigma_{X}=\sqrt{npq}=\sqrt{9*0.3*0.7}=1.3748$