X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X is known to be
10/3. What would be the probability that X assumes at most the value 2?
(b) 17/81.
(c) 47/243
(d) 46/243.
(a) 16/81.
Answers
Given : X is a binomial variable such that 2 P(X = 2) = P(X = 3) and mean of X is known to be 10/3
To find : What would be the probability that X assumes at most the value 2
Solution:
mean = 10/3
Mean = np = 10/3
P(X) = ⁿCₓpˣ(1-p)ⁿ⁻ˣ
=> P(2) = ⁿC₂p²(1-p)ⁿ⁻²
P(3) = ⁿC₃p³(1-p)ⁿ⁻³
P(3) = 2 P(2)
=> ⁿC₃p³(1-p)ⁿ⁻³ = 2 ⁿC₂p²(1-p)ⁿ⁻²
=> p/3!(n-3)! = (2 /2!(n-2)! )(1 - p)
=> p/6 = (1-p)/(n - 2)
=> np - 2p = 6 - 6p
=> 4p = 6 - np
=> 4p = 6 - 10/3
=> 4p = 8/3
=> p = 2/3
np = 10/3 => n = 5
probability that X assumes at most the value 2 = P(0) + P(1) + P(2)
= ⁵C₀(2/3)⁰(1/3)⁵ + ⁵C₁(2/3)¹(1/3)⁴ + ⁵C₂(2/3)²(1/3)³
= 1/243 + 10/243 + 40/243
= 51/243
= 17/81
17/81 is the probability that X assumes at most the value 2
Learn More:
The mean and variance of binomial distribution are 4 and 4/3 ...
https://brainly.in/question/3037214
A study found that 25% of car owners in UK had their cars washed ...
https://brainly.in/question/17355285
If a random variable X follows binomial distribution with mean as 5 ...
https://brainly.in/question/16640508