Find the median of the random variable X with the...

Find the median of the random variable $X$ with the probability density function defined on the indicated interval $I$. The median of $X$ is defined to be the number $m$ such that $P(X \leq m)=\frac{1}{2} .$ Observe that half of the $X$ -values lie below $m$ and the other half lie above $m$. $$f(x)=\frac{3}{16} \sqrt{x} ;[0,4]$$

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Key Concepts

Solving for the Median Using Integration

To find the median, one sets up an equation by integrating the pdf from the lower bound of the domain to an unknown m and equates this integral to 1/2. Solving this equation yields the median, ensuring that half of the probability lies below and half above this value.

Median of a Continuous Random Variable

The median is defined as the value m such that the cumulative probability up to m is one-half, meaning that there is a 50% chance that the variable will be below m and a 50% chance that it will be above m. It is found by solving the equation F(m) = 1/2, where F is the CDF.

Cumulative Distribution Function (CDF)

The cumulative distribution function gives the probability that a random variable is less than or equal to a specific value. It is obtained by integrating the pdf from the lower limit of the variable’s domain up to any point x. The CDF is a central tool in probability theory for evaluating probabilities and quantiles.

Probability Density Function (pdf)

A probability density function describes the relative likelihood for a continuous random variable to take on a given value. It must be nonnegative over its interval and integrate to one over its entire domain. The pdf provides the foundation for calculating probabilities by integration over specific intervals.

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