The random variable x is known to be uniformly distributed...

The random variable $x$ is known to be uniformly distributed between 10 and 20 .
a. Show the graph of the probability density function.
b. Compute $P(x<15)$
c. Compute $P(12 \leq x \leq 18)$
d. Compute $E(x)$
e. Compute Var $(x)$

Key Concepts

Variance

Variance is a statistical measure that quantifies the spread or dispersion of a random variable around its mean. For continuous distributions, it is calculated by determining the expected value of the squared deviation of the variable from its mean. In a uniform distribution, the variance can be derived using a standard formula, reflecting the degree to which values differ from the central value.

Expected Value

The expected value of a random variable is a measure of its central tendency, representing the mean of the distribution. For continuous distributions, it is computed as the integral of the product of the variable and its PDF over the entire range. In the context of uniform distributions, the expected value is typically found as the midpoint of the interval.

Probability Calculation in Continuous Distributions

For continuous random variables, the probability that the variable lies within a specific interval is calculated by integrating the PDF over that interval. This concept is crucial because it forms the basis for solving problems related to finding the likelihood of events, whether by direct integration or by utilizing properties of the PDF.

Probability Density Function (PDF)

The probability density function (PDF) of a continuous random variable describes the relative likelihood of the variable taking on a given value. For a uniform distribution, the PDF is a constant value over the interval of interest, which ensures that the total area under the curve equals one. Understanding the PDF is essential for computing probabilities and for visual representation of the distribution.

Uniform Distribution

A uniform distribution is a type of probability distribution in which all outcomes within a specified interval are equally likely. In continuous cases, this means that the probability density is constant over the interval, and zero elsewhere. This concept is fundamental in probability theory because it represents a state of maximum uncertainty when no outcome is favored over another within the range.

Graphical Representation of a PDF

Graphing a probability density function involves plotting the function for the variable, where the x-axis represents possible values and the y-axis shows the density. In the case of a uniform distribution, the graph is a horizontal line over the interval of interest which illustrates that every outcome in that interval is equally probable.

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