Question

Let X be a binomial random variable such that E[X] = 2 and Var (X) = 1.6. Find

(a) P{X > 2}

(b) P(XS 8)

(c) P(X= 14)​

Answer 1

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• p(x > 2) = p(x) = p(-2)

Answer 2

Answer:

X is a Binomial Random Variable. This means that the probability of X assuming the value r is

P(X=r)=(Nr)pr(1−p)N−r(1)

where p is the probability of the event occurring per try, and r is number of occurrences of the event in N tries.

We are given that 2⋅P(X=2)=P(X=3) , and np=103 . Thus, we can see that

2(N2)p2(1−p)N−2=(N3)p3(1−p)N−3(2)

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X is a binomial variable with probability of assuming a value x is p, then P(X=x)=C(n,x)px(1−p)n−x . It is required to find the probability that X assumes at most the value 2 which means it is required to find P(X=0)+P(X=1)+P(X=2) .

It is given 2P(X=2)=P(X=3)

⇒2C(n,2)p2(1−p)n−2=C(n,3)p3(1−p)n−3

⇒2∗n!2!(n−2)!(1−p)=n!3!(n−3)!p

⇒6(1−p)=(n−2)p

⇒6−6p=np−2p

⇒4p=6−np=6−10/3=8/3

⇒p=23 and q=1−p=13

Again, np=103⇒n=5

So, the required probability is

P(X=0)+P(X=1)+P(X=2)

=C(5,0)q5+C(5,1)pq4+C(5,2)p2q3

=135+5∗23∗134+10∗232∗133

=1+10+4035=51243=1781