Question

question 9
1 pts
the scheduled commuting time on the silver star rail road from trenton, nj to new york city is 68 minutes. suppose that the actual commuting time is uniformly distributed between 56 and 72 minutes. what is the probability that train will arrive in new york late? (that is, the trip takes longer than the scheduled commuting time.)
0.250
0.235
0.063
1.063
0.750
4.000
question 10
1 pts
the scheduled commuting time on amtrak from washington dc to baltimore, md is 40 minutes. suppose that the actual commuting time is uniformly distributed between 36 and 48 minutes. what is the expected commuting time that train will have?
40 minutes
12 minutes
36 minutes
6 minutes
42 minutes
48 minutes

Explanation:

Question 9

Step1: Recall Uniform Distribution Probability Formula

For a uniform distribution on interval \([a, b]\), the probability density function is \(f(x)=\frac{1}{b - a}\) for \(a\leq x\leq b\), and the probability \(P(X > c)\) (where \(c\) is a value in \([a, b]\)) is calculated as \(P(X > c)=\frac{b - c}{b - a}\) (since the area under the uniform distribution curve for \(x > c\) is a rectangle with width \(b - c\) and height \(\frac{1}{b - a}\)).

Here, the actual commuting time \(X\) is uniformly distributed between \(a = 56\) minutes and \(b = 72\) minutes. The scheduled time is \(c = 68\) minutes. We need to find \(P(X > 68)\).

Step2: Identify Values and Apply Formula

We have \(a = 56\), \(b = 72\), \(c = 68\).

First, calculate \(b - c\): \(72 - 68 = 4\).

Then, calculate \(b - a\): \(72 - 56 = 16\).

Now, apply the formula for probability in uniform distribution: \(P(X > 68)=\frac{b - c}{b - a}=\frac{4}{16}=\frac{1}{4}=0.250\).

Step1: Recall Expected Value Formula for Uniform Distribution

For a uniform distribution on the interval \([a, b]\), the expected value (mean) \(E(X)\) is given by the formula \(E(X)=\frac{a + b}{2}\).

Here, the actual commuting time \(X\) is uniformly distributed between \(a = 36\) minutes and \(b = 48\) minutes. We need to find the expected commuting time.

Step2: Identify Values and Apply Formula

We have \(a = 36\) and \(b = 48\).

Substitute these values into the formula: \(E(X)=\frac{36 + 48}{2}=\frac{84}{2}=42\) minutes.

Answer:

0.250

Question 10