Advertisements
Advertisements
Question
In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the mean and variance of the random variable
Solution
n = 5, X ~ B(n, p)
Given, P(X = 1) = 0.4096
P(X = 2) = 0.2048
P(X = x) = ⁿCx px qn-x, x = 0, 1, 2, …. n
⇒ 5C1 pq4 = 0.4096
⇒ 5C2 p2q3 = 0.2048
5pq4 = 0.4096 .......(1)
10p2q3 = 0.2048 .......(2)
(1) divided by (2),
⇒ `(5"pq"^4)/(10"p"^2"q"^3)` = 2
`"p"/"q"` = 4
q = 4p
q = 4(1 – q)
q = 4 – 4q
5q = 4
q = `4/5`
p = 1 – q = `1/5`
Mean = np
= `5 xx 1/5` = 1
Variance = npq
= `1 xx 4/5 = 4/5`
Distribution
P(X = x) = `""^5"C"_x (1/5)^x (4/5)^(5-x)`
x = 0, 1, 2, 3, 4, 5
APPEARS IN
RELATED QUESTIONS
Compute P(X = k) for the binomial distribution, B(n, p) where
n = 10, p = `1/5`, k = 4
The probability that Mr.Q hits a target at any trial is `1/4`. Suppose he tries at the target 10 times. Find the probability that he hits the target exactly 4 times
The probability that Mr.Q hits a target at any trial is `1/4`. Suppose he tries at the target 10 times. Find the probability that he hits the target at least one time
Using binomial distribution find the mean and variance of X for the following experiments.
A fair coin is tossed 100 times, and X denote the number of heads
Using binomial distribution find the mean and variance of X for the following experiments.
A fair die is tossed 240 times and X denotes the number of times that four appeared
A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be at least one defective item
A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be exactly two defective items?
If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights at least 11 will have a useful life of at least 600 hours
The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find the probability mass function
The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find P(X = 3)
The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find P(X ≥ 2)
If X ~ B(n, p) such that 4P(X = 4) = P(x = 2) and n = 6. Find the distribution, mean and standard deviation of X
Choose the correct alternative:
Let X have a Bernoulli distribution with a mean of 0.4, then the variance of (2X – 3) is
