The proportion of time Y that an industrial robot is in...

The proportion of time $Y$ that an industrial robot is in operation during a 40 -hour week is a random variable with probability density function $$f(y)=\left\{\begin{array}{ll} 2 y, & 0 \leq y \leq 1 \\ 0, & \text { elsewhere } \end{array}\right.$$ a. Find $E(Y)$ and $V(Y)$ b. For the robot under study, the profit $X$ for a week is given by $X=200 Y-60$. Find $E(X)$ and $V(X)$ c. Find an interval in which the profit should lie for at least $75 \%$ of the weeks that the robot is in use.

The proportion of time $Y$ that an industrial robot is in operation during a 40 -hour week is a random variable with probability density function
$$f(y)=\left\{\begin{array}{ll}
2 y, & 0 \leq y \leq 1 \\
0, & \text { elsewhere }
\end{array}\right.$$
a. Find $E(Y)$ and $V(Y)$
b. For the robot under study, the profit $X$ for a week is given by $X=200 Y-60$. Find $E(X)$ and $V(X)$
c. Find an interval in which the profit should lie for at least $75 \%$ of the weeks that the robot is in use.

Key Concepts

Chebyshev's Inequality

Chebyshev's inequality is a statistical tool that provides a bound on the probability that a random variable deviates from its mean by a specified number of standard deviations. It applies to any distribution with finite mean and variance, ensuring that at least a certain proportion of the distribution's values lie within a chosen interval around the mean, regardless of the distribution's shape.

Expected Value

The expected value of a continuous random variable is a measure of central tendency that is computed by integrating the product of the variable and its probability density function over its domain. This concept provides the mean or average value that the random variable takes, forming a baseline for further statistical analysis.

Variance

Variance is a measure of the spread or dispersion of a random variable around its mean. For continuous random variables, it is calculated as the expected value of the squared deviation from the mean, commonly found by computing the difference between the expected value of the square of the variable and the square of the expected value.

Linear Transformations of Random Variables

Linear transformations involve converting a random variable using a function of the form aY + b, where a and b are constants. In this process, the expectation is transformed by scaling and shifting (E[aY + b] = aE[Y] + b) and the variance is affected only by scaling (V[aY + b] = a²V[Y]), preserving the proportionality of variability.

Transcript

00:01 Here on this problem, we are dealing with a continuous random variable with probability distribution function given by 2y, when y is between 0 and 1.

00:18 Now, first here we'd like to find the expected value.

00:25 Now, the expected value of y is going to be the integral from 0 to 1 of y times f of y, the y.

00:33 In order to find the expected value of a random variable, if it is continuous, we take the integral of y times f of y.

00:42 Now here, this would be the integral from 0 to 1 of y times 2y, d .y.

00:55 And so this is the integral from 0 to 1 of 2 y squared, d .y.

01:04 Now, integrating gives us 2 thirds y cubed from y equals 0 to 1.

01:12 And so this gives us 2 over 3.

01:17 And so the expected value of y is 2 over 3.

01:22 Now we also want the variance.

01:23 Now before we can find the variance, we need to find the expected value of y squared.

01:30 And so this is the integral from 0 to 1 of y squared times f of y, d y.

01:38 So this is the integral from 0 to 1 of y squared times 2y, dy, which is the integral from 0 to 1 to 2y cubed, dy.

01:55 Now iterating this gives us 1⁄2 .y to the 4th from y to 1 .4.

02:04 Go 0 to 1 and so it gives us 1 half until e of y squared is 1 half.

02:12 Now for the variance, though, which is what we were really after, the variance is equal to e of y squared minus e of y squared...

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