Question

question 6
1 pts
the random variable x is known to be uniformly distributed between 3.62 and 8.17. compute ( p(5.64 < x < 7.82) ).
0.367
2.087
0.479
0.459
0.220
0.370
question 7
1 pts
the random variable x is known to be uniformly distributed between 3.19 and 9.58. compute the probability that x is exactly 6.
0.939
0.440
0
0.940
0.061
0.560
question 8
1 pts

Explanation:

Question 6

Step1: Recall Uniform Distribution Probability Formula

For a uniform distribution between \(a\) and \(b\) (\(a < b\)), the probability density function is \(f(x)=\frac{1}{b - a}\) for \(a\leq x\leq b\), and the probability \(P(c < x < d)\) (where \(a\leq c < d\leq b\)) is given by \(P(c < x < d)=(d - c)\times\frac{1}{b - a}\). Here, \(a = 3.62\), \(b = 8.17\), \(c = 5.64\), \(d = 7.82\).

Step2: Calculate \(b - a\)

\(b - a=8.17 - 3.62 = 4.55\)

Step3: Calculate \(d - c\)

\(d - c=7.82 - 5.64 = 2.18\)

Step4: Compute the Probability

Using the formula \(P(5.64 < x < 7.82)=\frac{d - c}{b - a}=\frac{2.18}{4.55}\approx0.479\)

Step1: Recall Property of Continuous Distributions

In a continuous probability distribution (like the uniform distribution), the probability of a random variable taking on a single exact value is \(0\). This is because the area under a single point in a continuous distribution (which is represented by a line with no width) is \(0\). For a uniform distribution between \(a = 3.19\) and \(b = 9.58\), \(P(x = 6)\) is the probability of \(x\) being exactly \(6\), which is a single point in the continuous range.

Answer:

0.479

Question 7