Demand The daily demand x for a certain product (in hundreds...

Demand The daily demand $x$ for a certain product (in hundreds of pounds) is a random variable with the probability density function $f(x)=\frac{1}{36} x(6-x),[0,6]$ (a) Determine the expected value and the standard deviation of the demand. (b) Determine the median of the random variable. (c) Find the probability that $x$ is within one standard deviation of the mean.

Demand The daily demand $x$ for a certain product (in hundreds of pounds) is a random variable with the probability density function $f(x)=\frac{1}{36} x(6-x),[0,6]$
(a) Determine the expected value and the standard deviation of the demand.
(b) Determine the median of the random variable.
(c) Find the probability that $x$ is within one standard deviation of the mean.

Key Concepts

Continuous Probability Distribution

A continuous probability distribution describes the probabilities of the outcomes of a continuous random variable. Unlike a discrete random variable that takes countable values, a continuous random variable can take on any value within an interval. Its behavior is defined through a probability density function (pdf) where the probability of the variable falling within a particular interval is calculated by integrating the pdf over that interval.

Expected Value

The expected value, or mean, of a continuous random variable represents a measure of its central tendency. It is calculated by taking the integral of the product of the variable and its pdf over the entire range of possible values. This operation provides a weighted average of all possible outcomes, where the weights are the probabilities of those outcomes.

Variance and Standard Deviation

Variance quantifies the spread or dispersion of the probability distribution by measuring the average of the squared deviations from the expected value. The standard deviation is the square root of the variance and provides a scale of dispersion that is in the same units as the random variable. Both are key measures for understanding the reliability and variability of the distribution.

Median

The median of a continuous distribution is the value that divides the probability mass into two equal halves. It is defined as the point where the integral of the pdf from the lower bound up to that point equals one-half. The median provides another measure of central tendency that is less sensitive to extreme values compared to the mean.

Probability Calculation using Integration

In continuous distributions, the probability that a random variable falls within a specific interval is computed by integrating the probability density function over that interval. This method is fundamental for addressing various probability queries, such as determining the probability that the value lies within a range defined by certain criteria like being within one standard deviation of the mean.

Please give Ace some feedback