You have been asked to determine the probability that the...

You have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of $$\$ 2,000$$. The total number of units sold is a normally distributed random variable with a mean of 400 and a variance of 900 , $X \sim N(400,900)$. The selling price per unit is $$\$ 10$$. The total number of units produced is a normally distributed random variable with a mean of 400 and a variance of $1,600, Y \sim N(400,1,600)$. The variable production cost is $$\$ 4$$ per unit. Production and sales have a positive correlation of 0.50 .

Key Concepts

Probability Calculation

Probability calculation in this context involves determining the likelihood that a certain financial metric, such as the contribution margin, exceeds a given threshold like fixed costs. This requires understanding the underlying probability distributions of the revenue and costs, and applying statistical methods to evaluate the chance that the overall result falls within a desired range. This approach is vital for risk assessment and informed decision making in financial planning.

Contribution Margin Analysis

Contribution margin analysis is a key concept in cost accounting that involves assessing the difference between a product's sales revenue and its variable costs. This margin is critical for covering fixed costs and achieving profitability. By comparing the contribution margin to fixed costs, one can evaluate whether a product line is generating enough revenue to break even or deliver profit.

Linear Combination of Random Variables

The linear combination of random variables involves combining variables using constant coefficients. When the variables are normally distributed—even if they are correlated—the linear combination will also be normally distributed. This property is particularly useful for analyzing functions of random variables, such as calculating overall costs or revenues, because the resulting distribution can be used to determine probabilities of specific outcomes through its mean and variance.

Bivariate Normal Distribution

A bivariate normal distribution describes the joint behavior of two normally distributed random variables when they are possibly correlated. This concept is essential when dealing with two interrelated variables since it takes into account not only the individual properties of each variable but also the degree to which they move together, expressed by the correlation coefficient. Understanding this helps in determining the joint likelihoods and probabilities involving both variables.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, symmetric about the mean. It is defined by two parameters—the mean and the variance—which describe the central tendency and the spread of the distribution, respectively. This concept is fundamental in probability and statistics as many natural phenomena and processes can be approximated by a normal distribution, and many statistical methods assume normality.

Transcript

00:01 Incorrect, yep, 400 and 900, and the other one is 400 and 1600.

00:06 And for why, this mean of why is 400 and the variance is 1 ,600.

00:15 And so we have our margin that they're talking about being equivalent to the cost and the amount that each of these are sold for is $10.

00:26 And each of these to make them, it cost them $4.

00:30 So if we take 10 times the number sold minus four times the number made, then that will end up giving us our contribution margin.

00:42 And we want to find what the mean of m is and what the standard deviation or the variance of m is.

00:50 And so our mean is going to be easy.

00:53 That's going to be that 10 times 400 minus the 4 times the 400.

01:02 Which is going to be the 4 ,000 minus the 1600, which means the mean of m is equal to 2 ,400.

01:14 And now our variance, we're going to have, oh, we know one more thing.

01:18 We know that the correlation between those sold and made is equal to 0 .5, positive 0 .5.

01:24 So now let's find our variance for m and for that margin.

01:30 And then we have to answer probability question for that.

01:33 So we have to take 100 times the variance of the first variable, which is 900, plus the value squared, 16 times the variance, which is 1 ,600.

01:46 And then because of this being a minus, we'll put minus.

01:49 And we have to take 2 times a times b, which is 40, times the correlation.

01:57 So this is times, continue that over times the correlation coefficient, times the standard deviation of the first.

02:03 Variable which would be 30 and then times the standard deviation of the second variable which would be 40 and so let's get that variance so i have 100 times 900 plus 16 times 1 ,600 minus 2 times 40 times and 0 .5 and then times in this 30 times 40 is going to be 1200 and so we get our variance of m to be 67 ,600...

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