I have a test on this tomorrow and I swear he didn't teach this and the book's not helping either, so maybe u can:
A hwy engineer knows that his crew can lay 5 miles of hwy on a clear day, 2 miles on a rainy day, and only 1 mile on a snowy day. Suppose that the probabilities are as follows:
Outcome prob/random variable: clear .6/5, rain .3/2, snow .1/1
Find the mean (expected value) and the variance
Can anyone at least tell me where to start????
X P(X) x*P(X) x^2*P(X)
5 0.6 3 15
2 0.3 0.6 1.2
1 0.1 0.1 0.1
Mean = sum x*P(X)=3.7
sum (x^2 * P(X)) = 16.3
Variance = (x^2 * P(X)) - Mean^2
=16.3 - (3.7)^2 = 2.61
Mean = 3.7 miles and variance = 2.61 miles square
To find the mean (expected value) and variance, you'll need to multiply each outcome by its probability and then sum them up. Here's how you can start:
1. Assign variables:
Let X represent the number of miles of highway laid.
Let P(X = 5) represent the probability of the crew laying 5 miles on a clear day.
Let P(X = 2) represent the probability of the crew laying 2 miles on a rainy day.
Let P(X = 1) represent the probability of the crew laying 1 mile on a snowy day.
2. Calculate the mean (expected value):
The mean (expected value) is calculated as the sum of the products of each outcome and its probability. In this case, it would be:
Mean (E[X]) = 5 * P(X = 5) + 2 * P(X = 2) + 1 * P(X = 1)
Substitute the provided probabilities from the question into the equation and calculate:
Mean (E[X]) = 5 * 0.6/5 + 2 * 0.3/2 + 1 * 0.1/1
Mean (E[X]) = 0.6 + 0.3 + 0.1
Mean (E[X]) = 1
So, the mean (expected value) is 1.
3. Calculate the variance:
To calculate the variance, you'll need to subtract the mean from each outcome, square the result, multiply it by the probability, and then sum them up. Use the formula:
Variance (Var(X)) = (5 - E[X])^2 * P(X = 5) + (2 - E[X])^2 * P(X = 2) + (1 - E[X])^2 * P(X = 1)
Substitute the provided probabilities and the mean value into the equation:
Variance (Var(X)) = (5 - 1)^2 * 0.6/5 + (2 - 1)^2 * 0.3/2 + (1 - 1)^2 * 0.1/1
Variance (Var(X)) = 4 * 0.6/5 + 1 * 0.3/2 + 0
Variance (Var(X)) = 0.48 + 0.15 + 0
Variance (Var(X)) = 0.63
So, the variance is 0.63.