Math Problem Statement
The mean and variance of a Binomial Variate X with parameters n and p are 16 and 0.8 respectively. Find (i) P(X = 0) (ii) P(X ≥ 2).
Solution
Given that is a binomial random variable with parameters (number of trials) and (probability of success), the mean () and variance () of are given by:
Step 1: Determine and
From the mean equation:
From the variance equation:
Substitute into Equation 2:
Solve for :
Now, substitute into Equation 1 to find :
Since must be an integer (as it's the number of trials), we round to 17.
Step 2: Find the required probabilities
(i)
(ii)
First, calculate :
Now, calculate :
So, .
Final Answers:
Do you want details on any specific steps or have any questions?
Here are five related questions to consider:
- How would you compute the probability for the same binomial distribution?
- What would be the effect on the mean and variance if the value of was 0.9 instead of 0.95?
- How would the probabilities change if was a different integer, say 20?
- What is the cumulative probability ?
- How does the binomial distribution behave as increases with a fixed ?
Tip: In binomial distributions, the variance is maximized when , and it decreases as approaches 0 or 1.
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Probability
Formulas
Binomial probability formula
Mean and variance of binomial distribution
Theorems
-
Suitable Grade Level
Advanced High School
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