Probability and Statistics: Continuous Random Variables

Suppose that $f(x)$, the PDF function, is $\frac{x}{8}$ for $x$ between 3 and 5.

Determine the following probabilities: (a) $\Pr(X < 4)$, (b) $\Pr(4 < X < 5)$, (c) $\Pr(X < 3.5 \text{ or } X > 4.5)$.

If we have a uniform distribution from A = 2 to B = 6, what is the value of the probability density function $f(X)$?

Using the PDF of an exponential distribution $f(x) = \lambda e^{-\lambda x}$, calculate the probability that $X$ is less than a given value $x$.

Find the probability that all observations in a random sample from an exponential distribution with mean 3 have a value greater than 4.

Scores on an exam are normally distributed with a mean of 65 and a standard deviation of 9. We want to find the percent of scores satisfying a) $x < 54$, b) $x \ge 80$, c) $70 < x < 86$.

You love pizza, and your local pizza shop claims that their large is at least 16 inches or it's free. Over your expansive pizza eating career, you've found that their pizzas are normally distributed in size with a mean of 16.3 inches and a standard deviation of 0.2 inches.

Now you want to know: What is the probability of getting a free pizza? Also, what's the probability of getting a pizza that's over 16.5 inches? And, what's the probability of getting a pizza that's between 15.95 inches and 16.63 inches?

Illustrate a continuous distribution example using women's heights, explaining the probability density function (PDF) and cumulative distribution function (CDF).

Link the PDF and CDF by discussing how the gradient of the CDF corresponds to the PDF for continuous distributions.

What is the probability that the time between two customer arrivals is less than five minutes?

Find the probability that Z is less than -1.32 using a standard normal distribution table.

Find the probability that Z is greater than -1.32.

Find the probability that Z is between -0.21 and 0.85.

Scores on an exam are normally distributed with a mean (mu) of 70 and a standard deviation (sigma) of 7. We want to find the probability of obtaining scores in the intervals listed in a) to d) below. In a), we want the probability that a score X is less than 84. ... In part c) we want the probability that X is between 60 and 74.

What is the proportion of all people that have heights greater than 170 centimeters?

What is the probability a randomly selected order has at least 760 calories?

Every day the bus is uniformly late between 2 and 10 minutes. On any given day, how long can you expect to wait and what is the standard deviation of the wait time? If the bus is late more than seven minutes, you'll be late to work, so what is the probability that you'll be late on any given day?

For a bus that arrives every hour but with an unknown departure time, answer the following questions: (1) What is the probability density function for this situation? (2) How do we draw a continuous distribution graph for this situation? (3) What is the likelihood of having to wait between 5 and 20 minutes for the bus? (4) What are the mean and standard deviation for this situation?

What proportion of students scored less than 49 on the exam if the scores are normally distributed with a mean of 60 and a standard deviation of 10?

What proportion of students are between 5.81 feet and 6.3 feet tall if the heights are normally distributed with a mean height of 5.5 feet and a standard deviation of 0.5 feet?