To determine the center and shape of the distribution for a binomial random variable with parameters [tex]\( n = 5 \)[/tex] and [tex]\( p = 0.5 \)[/tex], we need to first understand a few key concepts about binomial distributions.
### Step 1: Understanding the Binomial Distribution
A binomial distribution is defined by two parameters:
- [tex]\( n \)[/tex]: the number of trials.
- [tex]\( p \)[/tex]: the probability of success on a single trial.
The binomial random variable [tex]\( X \)[/tex] counts the number of successes in [tex]\( n \)[/tex] independent trials with success probability [tex]\( p \)[/tex].
### Step 2: Calculating the Center (Mean) of the Distribution
The mean (or expected value) of a binomial distribution is given by the formula:
[tex]\[ \text{Mean} = \mu = n \times p \][/tex]
For [tex]\( n = 5 \)[/tex] and [tex]\( p = 0.5 \)[/tex]:
[tex]\[ \mu = 5 \times 0.5 = 2.5 \][/tex]
Thus, the center of the distribution is 2.5.
### Step 3: Understanding the Shape of the Distribution
The shape of the binomial distribution depends largely on the value of [tex]\( p \)[/tex]:
- If [tex]\( p = 0.5 \)[/tex], the distribution is symmetric because the number of successes is just as likely to be more than the mean as it is to be less than the mean.
- If [tex]\( p \)[/tex] is not equal to 0.5, the distribution is typically skewed. If [tex]\( p \)[/tex] is closer to 0 or 1, the skewness is more pronounced.
With [tex]\( p = 0.5 \)[/tex] in our case, the distribution is symmetric because each outcome (success or failure) is equally likely.
### Conclusion
Based on the calculations and understanding of the binomial distribution with [tex]\( n = 5 \)[/tex] and [tex]\( p = 0.5 \)[/tex]:
- The center (mean) of the distribution is 2.5.
- The shape of the distribution is symmetric.
So, the correct answer is:
Center: 2.5, Shape: symmetric.
