Question

QUESTIONS 1, 2 AND 3 ARE BASED ON THE FOLLOWING INFORMATION.

Around elections time the number of daily fake news posts about politicians follows the following discrete probability distribution.

x 0 1 2 3 4
P (X = x) 0.1 0.15 0.2 0.25 ?

Let X be the number of daily fake news posts.

Question 1

Which one of the following statements is incorrect?

(1) ∑_ P(x) = 1
(2) P(X = 4) = 0
(3) P(X ≤ 4) = 1
(4) P(X ≥ 4) = 0.3
(5) None of the above.

Question 2

What is the expected number of daily fake news posts about politicians around elections time?

(1) 5.3
(2) 4
(3) 2.5
(4) 2.1
(5) 1.3


Question 3

What is the standard deviation of daily fake news posts about politicians around election time?

(1) 1.32
(2) 1.75
(3) 2.5
(4) 1
(5) None of the above

Answer 1

since $\sum\limits_{x=0}^{4}P(X=x) =1$

therefore $P(X=4)=1-(0.1+0.15+0.2+0.25)=0.3$

therefore ,

In Question 01,

answer is (2)

$P(X=4)=0$ is incorrect and correct one is $P(X=4)=0.3$

In Question 02,

expected number E[X],

$E[X]=\sum\limits_{i=0}^{4}xP(X=x)\\ E[X]=0*0.1+1*0.15+2*0.2\\\hspace {4 em}+3*0.25+4*0.3\\ E[X]=2.5$

answer is (3)

In Question 03,

standard deviation =$\sqrt{variance}=\sqrt{V(X)}$

$V(X)=E[X^2]-E[X]^2\\ E[X^2]=\sum\limits_{i=0}^{4}x^2P(X=x)\\ E[X^2]=0^2*0.1+1^2*0.15+2^2*0.2\\\hspace {4 em}+3^2*0.25+4^2*0.3\\ E[X^2]=8\\ V(X)=8-2.5^2=1.75\\ standard\ deviation=\sqrt{1.75}=1.322$

answer is (1)