Which statement about a uniform probability distribution defined on a given interval is true?
A) The mean is always 0.
B) The mean is always 1.
C) The standard deviation is always 1.
D) The mean is the midpoint of the interval.
E) The standard deviation is equal to the width of the interval.
I believe D, because the mean of X~U(a,b) is E(x) =(a+b)/2
To determine the correct statement about a uniform probability distribution defined on a given interval, we need to understand what a uniform probability distribution is.
A uniform probability distribution is a continuous probability distribution where all outcomes in a given interval are equally likely. This means that the probability of any particular value occurring within the interval is equal.
Now, let's analyze each statement:
A) The mean is always 0.
To find the mean of a uniform probability distribution, we need to calculate the average of the values within the interval. Since the distribution is uniform, all values are equally likely, meaning they contribute equally to the mean. Therefore, the mean is not always 0. This statement is false.
B) The mean is always 1.
Similar to statement A, this statement is false. The mean of a uniform probability distribution is not always 1. The mean depends on the interval and is not fixed.
C) The standard deviation is always 1.
To find the standard deviation of a uniform probability distribution, we need to calculate the square root of the variance. The variance is calculated by taking the difference between each value and the mean, squaring those differences, and then averaging them. In a uniform distribution, the values have an equal distance from the mean, resulting in a variance that is not always 1. Therefore, this statement is false.
D) The mean is the midpoint of the interval.
This statement is true. In a uniform probability distribution, the mean is the midpoint of the interval. This is because the distribution is symmetrical, and the midpoint represents the balance point of the values.
E) The standard deviation is equal to the width of the interval.
This statement is false. The standard deviation is not equal to the width of the interval in a uniform distribution. The standard deviation depends on the spread of the values within the interval, which can vary.
In conclusion, the correct statement about a uniform probability distribution defined on a given interval is D) The mean is the midpoint of the interval.