Textbook Question
"Consider a binomial probability distribution with parameters n = 5 and p = 0.2.
a. Construct a binomial probability distribution with these parameters."
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5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
Multiple Choice
Which of the following graphs best represents the probability mass function of a binomial distribution with and ?
A
A histogram with bars peaking at and decreasing towards both and , showing a left-skewed distribution.
B
A histogram with bars peaking at and symmetric about , showing a bell-shaped distribution.
C
A histogram with all bars of equal height from to , showing a uniform distribution.
D
A histogram with bars starting at , peaking around , and then decreasing towards , showing a right-skewed distribution.
Verified step by step guidance1
Recall that the binomial distribution models the number of successes in \( n \) independent Bernoulli trials, each with success probability \( p \). Here, \( n = 20 \) and \( p = 0.25 \).
Calculate the mean (expected value) of the binomial distribution using the formula:
\[ \mu = n \times p \]
This gives the center or peak location of the distribution.
Understand the shape of the binomial distribution based on \( p \):
- If \( p = 0.5 \), the distribution is symmetric and bell-shaped.
- If \( p < 0.5 \), the distribution is right-skewed (peak near 0).
- If \( p > 0.5 \), the distribution is left-skewed (peak near \( n \)).
Since \( p = 0.25 < 0.5 \), expect the distribution to be right-skewed, with the highest probability mass near \( k = 5 \) (the mean), and probabilities decreasing as \( k \) approaches 20.
Compare the given histogram descriptions to this expected shape: the correct graph should show bars starting at 0, peaking around 5, and then decreasing towards 20, indicating a right-skewed distribution.
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