Suppose that X represents the daily sales of a particular...

Suppose that $X$ represents the daily sales of a particular product, and that the probability distribution of $X$ is as follows: $$ \begin{array}{cc} \hline x & P(X=x) \\ \hline 7,000 & .05 \\ 7,500 & .20 \\ 8,000 & .35 \\ 8,500 & .19 \\ 9,000 & .12 \\ 9,500 & .08 \\ 10,000 & .01 \\ \hline \end{array} $$ Find the expectation and the variance of daily sales. [Hint: To find the variance, it is easiest to first find the variance of a different variable, such as $Y=(X-7000) / 1000$. ] If the net profit resulting from sales of $X$ items can be given by $$ Z=5 X-38,000, $$ find the expectation and the variance of net profit, $Z$.

Suppose that $X$ represents the daily sales of a particular product, and that the probability distribution of $X$ is as follows:
$$
\begin{array}{cc}
\hline x & P(X=x) \\
\hline 7,000 & .05 \\
7,500 & .20 \\
8,000 & .35 \\
8,500 & .19 \\
9,000 & .12 \\
9,500 & .08 \\
10,000 & .01 \\
\hline
\end{array}
$$
Find the expectation and the variance of daily sales. [Hint: To find the variance, it is easiest to first find the variance of a different variable, such as $Y=(X-7000) / 1000$. ] If the net profit resulting from sales of $X$ items can be given by
$$
Z=5 X-38,000,
$$
find the expectation and the variance of net profit, $Z$.

Key Concepts

Expected Value

The expected value (or mean) of a random variable is a measure of its central or average value. It is calculated as the sum of all possible values weighted by their corresponding probabilities. The expected value provides a summary of the distribution of the random variable and is especially useful in understanding the long-run behavior of random processes.

Variance

Variance measures the dispersion or spread of the random variable's values around its mean. It is computed as the expected value of the squared deviation of the variable from its mean. Variance is a key concept in assessing the reliability and risk associated with random outcomes, as well as in further statistical analysis.

Discrete Random Variable

A discrete random variable is one that takes on a countable set of possible values, each with an associated probability. In probability theory, discrete random variables are characterized by a probability mass function (PMF), which lists the probabilities of each outcome. Understanding these variables is crucial for calculating measures such as expectation and variance, which are central to statistical analysis.

Linear Transformation of Random Variables

A linear transformation of a random variable involves scaling and/or shifting its values. This property allows for the straightforward computation of the expected value and variance of the transformed variable. Specifically, the linearity of expectation means that the expected value of a transformed variable is the transformation applied to the original expected value, while the variance is affected by the square of the scaling factor. This concept is fundamental in statistical modeling and analysis.

Transcript

Please give Ace some feedback