For certain ore samples, the proportion Y of impurities per...

For certain ore samples, the proportion $Y$ of impurities per sample is a random variable with density function given in Exercise $4.21 .$ The dollar value of each sample is $W=5-.5 Y$. Find the mean and variance of $W$

Key Concepts

Linear Transformation of Random Variables

A linear transformation involves changing a random variable through operations of scaling and shifting, typically expressed as W = a + bY. This transformation affects the expected value and variance in predictable ways: the mean becomes a + b*E(Y) and the variance becomes b²*Var(Y). These properties simplify the analysis of transformed variables by allowing direct calculation of their statistical properties from the original variable.

Probability Density Function

A probability density function (pdf) represents the distribution of a continuous random variable. It describes how the probability is distributed over the values of the variable, and this function must integrate to one over the support of the variable. The pdf is used to calculate expectations and probabilities by integrating over appropriate intervals.

Expected Value

The expected value, or mean, of a random variable provides a measure of the central tendency or the long-run average outcome of the variable. It is computed by integrating the product of the variable's value and its probability density function over all possible values, offering insight into where the outcomes tend to cluster.

Variance

Variance measures the spread or dispersion of a random variable's values around its mean. It quantifies how much the values of the variable deviate from the expected value by taking the average of the squared deviations. Variance is a crucial concept for understanding the reliability and variability inherent in the random process.

Transcript

00:01 Hello, this is prome 27.

00:03 We're talking about our samples.

00:07 Well, there are impurities actually.

00:10 So the first thing we need to know is the density, which is f of y, which is you go to 3 .5.

00:23 Y squared plus y.

00:26 And it's, it lives in between zero and one.

00:33 And if it's not in between zero, there then it's zero elsewhere so we're trying to find the expected value of w and the variance of w but before we get to that we need to find the expected value of y which is a proportion of impurities in our samples so it's defined right between zero and one and we multiply one by the density, as we have seen before.

01:21 Now, we need to take, we need to simplify, right? so we keep the same limit, 0 to 1.

01:30 3 .5.

01:32 Y cubed plus y squared, d .y.

01:38 Taking the integral, we'll get 3 .8s.

01:44 Y cubed.

01:47 Or y to the fourth, actually.

01:50 Y to the fourth plus y cubed divided by three and we're going to evaluate from zero to one simplifying we'll get 0 .708 so that's the expected value of the proportion of inquiries and our samples so now we'll figure out the expected value of the expected value to the integral from 0 to 1 of y squared multiplied by the density.

02:34 3 1, y squared plus y, d, y.

02:41 Simplifying, we'll get the integral from 0 to 1 of 3 .5, y to the 4th plus y cubed, which is equal to, well now we need to take the integral.

02:57 So it's going to be 3 tenths, y to the fifth, plus y to the fourth over four.

03:07 And we're going to evaluate this from 0 to 1.

03:11 Simplifying, we'll get 0 .55.

03:15 Now, we need to find the whole point of this is finding the variance of y.

03:21 So the variance of y is equal to the expected value of y squared minus the expected value...

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