A random variable X has a pdf f(x) defined over an interval...

A random variable $X$ has a pdf $f(x)$ defined over an interval $[a, b]$ where $b>a>0$ The equation that defines $f(x)$ is $f(x)=k$ a) Find $k$ in terms of $a$ and $b$ b) Also, in terms of $a$ and $b,$ find (i) the mean (ii) the median (iii) the variance. Note: This is known as the uniform distribution.

A random variable $X$ has a pdf $f(x)$ defined over an interval $[a, b]$ where $b>a>0$ The equation that defines $f(x)$ is $f(x)=k$
a) Find $k$ in terms of $a$ and $b$
b) Also, in terms of $a$ and $b,$ find
(i) the mean
(ii) the median
(iii) the variance.
Note: This is known as the uniform distribution.

Key Concepts

Variance

The variance measures the spread or dispersion of the random variable around the mean. In the context of a uniform distribution, the variance is determined by the square of the width of the interval divided by a factor (typically 12), quantifying how much the values of the random variable are spread out from the central mean.

Median

The median of a probability distribution is the value that divides the distribution into two equal halves, such that there is a 50% probability of the random variable being below or above this value. For a symmetric uniform distribution, the median is the same as the mean, being directly in the middle of the interval.

Uniform Distribution

The uniform distribution is a probability distribution in which all outcomes are equally likely over a given interval. In this distribution, the probability density function (pdf) is constant over the interval, meaning that every subinterval of the same length within the domain has an identical probability of occurrence.

Normalization of the Probability Density Function

The pdf of any continuous random variable must integrate to 1 over its entire range, which is known as the normalization condition. For a constant pdf over an interval, this condition is used to determine the constant value, ensuring that the total probability across the specified interval sums to one.

Mean

The mean, or expectation, of a random variable is a measure of the central tendency of the distribution. For a uniform distribution over an interval, the mean is calculated as the arithmetic average of the endpoints of the interval, reflecting the symmetric nature of the probability spread.

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