Combination of Random Variables: Golf Norb and Gary are...

Combination of Random Variables: Golf Norb and Gary are entered in a local golf tournament. Both have played the local course many times. Their scores are random variables with the following means and standard deviations. $$ \text { Norb, } x_{1}: \mu_{1}=115 ; \sigma_{1}=12 \quad \text { Gary, } x_{2}: \mu_{2}=100 ; \sigma_{2}=8 $$ In the tournament, Norb and Gary are not playing together, and we will assume their scores vary independently of each other. (a) The difference between their scores is $W=x_{1}-x_{2}$. Compute the mean, variance, and standard deviation for the random variable $W$. (b) The average of their scores is $W=0.5 x_{1}+0.5 x_{2}$. Compute the mean, variance, and standard deviation for the random variable $W$. (c) The tournament rules have a special handicap system for each player. For Norb, the handicap formula is $L=0.8 x_{1}-2 .$ Compute the mean, variance, and standard deviation for the random variable $L .$ (d) For Gary, the handicap formula is $L=0.95 x_{2}-5 .$ Compute the mean, variance, and standard deviation for the random variable $L .$

Combination of Random Variables: Golf Norb and Gary are entered in a local golf tournament. Both have played the local course many times. Their scores are random variables with the following means and standard deviations.
$$
\text { Norb, } x_{1}: \mu_{1}=115 ; \sigma_{1}=12 \quad \text { Gary, } x_{2}: \mu_{2}=100 ; \sigma_{2}=8
$$
In the tournament, Norb and Gary are not playing together, and we will assume their scores vary independently of each other.
(a) The difference between their scores is $W=x_{1}-x_{2}$. Compute the mean, variance, and standard deviation for the random variable $W$.
(b) The average of their scores is $W=0.5 x_{1}+0.5 x_{2}$. Compute the mean, variance, and standard deviation for the random variable $W$.
(c) The tournament rules have a special handicap system for each player. For Norb, the handicap formula is $L=0.8 x_{1}-2 .$ Compute the mean, variance, and standard deviation for the random variable $L .$
(d) For Gary, the handicap formula is $L=0.95 x_{2}-5 .$ Compute the mean, variance, and standard deviation for the random variable $L .$

Key Concepts

Linear Combination of Random Variables

A linear combination of random variables involves adding or subtracting multiple random variables after multiplying them by constants. The mean of the linear combination is the same linear combination of the means, and if the variables are independent, the variance is the sum of the squared constants times their variances.

Independence of Random Variables

When random variables are independent, the occurrence or outcome of one variable does not influence the other. This property simplifies calculations of variances for sums or differences, as there are no covariance terms to consider.

Standard Deviation

The standard deviation is the positive square root of the variance, giving a measure of dispersion in the same units as the random variable itself. It is widely used to understand the spread or variability around the mean.

Linear Transformation of Random Variables

A linear transformation scales and shifts a random variable using a formula of the form L = aX + b. This transformation changes the mean by scaling and adding a constant and changes the variance by the square of the scaling factor, while the constant term b does not affect the variance.

Expected Value (Mean)

The expected value of a random variable is its long-run average value, calculated as the weighted sum of all possible values, where the weights are their corresponding probabilities. This measure indicates the central tendency of the distribution.

Variance

Variance is a measure of how far a set of numbers (random variable outcomes) are spread out from their mean. It is computed as the expected value of the squared deviations from the mean, providing insight into the dispersion of the distribution.

Random Variables

A random variable is a numerical outcome of a random phenomenon. It can take various values according to a probability distribution and is fundamental in probability and statistics for modeling uncertainty.

Transcript

0:00 19.

00:02 When do creative people get their best ideas? usa today did a survey of 966 inventors all who hold a u .s.

00:08 Patent and obtain the following information.

00:10 So in the textbook, we're given that 290 inventors get their best ideas between 6 a .m.

00:16 And 12 noon.

00:17 135 inventors get their best ideas between 12 noon and 6 p .m., 319 between 6 p .m.

00:23 And 12 midnight, and 222 between 12 midnight and 6 a .m.

00:29 A.

00:34 To make the probability that an adventure has his best idea during each time interval.

00:39 So let's write down those time intervals.

00:40 Those time intervals were from 6 a .m.

00:42 To 12 noon, 12 noon to 6 p .m.

00:50 To 12 at midnight, and 12 midnight to 6 a .m.

00:56 So it also told us the number.

00:59 So 290, 135, 319, and 222.

01:06 So there's a total of 966 inventors.

01:10 We just simply have to find the ratio of each of these to figure out the probabilities.

01:16 So we're going to go 290 divided by 966 gives us 0 .30, 135 divided by 966.

01:28 We round that to 0 .14.

01:31 319 divided by 966 gives us 0 .33 and 222 divided by 966 gives us 0 .33 and 222 divided by 966...

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