5.13 Discrete and continuous random variables: homework  (Page 4/5)

A random number generator picks a number from 1 to 9 in a uniform manner.

• a $X\text{~}$
• b Graph the probability distribution.
• c $f(x)=$
• d $\mu =$
• e $\sigma =$
• f
• g $P(x>5\text{.}\text{67})=$
• h $P(x>5\mid x>3)=$
• i Find the 90th percentile.

The time (in minutes) until the next bus departs a major bus depot follows a distribution with $f(x)=\frac{1}{\text{20}}$ where $x$ goes from 25 to 45 minutes.

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d The distribution is ______________ (name of distribution). It is _____________ (discrete or continuous).
• e $\mu =$
• f $\sigma =$
• g Find the probability that the time is at most 30 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.
• h Find the probability that the time is between 30 and 40 minutes. Sketch and label a graph of the distribution. Shade the area of interest. Write the answer in a probability statement.
• i _________. State this in a probability statement (similar to g and h ), draw the picture, and find the probability.
• j Find the 90th percentile. This means that 90% of the time, the time is less than _____ minutes.
• k Find the 75th percentile. In a complete sentence, state what this means. (See j .)
• l Find the probability that the time is more than 40 minutes given (or knowing that) it is at least 30 minutes.

According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight. Let’s suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month. (Source: The McDougall Program for Maximum Weight Loss by John A. McDougall, M.D.)

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d $f(x)=$
• e $\mu =$
• f $\sigma =$
• g Find the probability that the individual lost more than 10 pounds in a month.
• h Suppose it is known that the individual lost more than 10 pounds in a month. Find the probability that he lost less than 12 pounds in the month.
• i __________. State this in a probability question (similar to g and h), draw the picture, and find the probability.

A subway train on the Red Line arrives every 8 minutes during rush hour. We are interested in the length of time a commuter must wait for a train to arrive. The time follows a uniform distribution.

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d $f(x)=$
• e $\mu =$
• f $\sigma =$
• g Find the probability that the commuter waits less than one minute.
• h Find the probability that the commuter waits between three and four minutes.
• i 60% of commuters wait more than how long for the train? State this in a probability question (similar to g and h ), draw the picture, and find the probability.

The age of a first grader on September 1 at Garden Elementary School is uniformly distributed from 5.8 to 6.8 years. We randomly select one first grader from the class.

• a Define the random variable. $X=$
• b $X\text{~}$
• c Graph the probability distribution.
• d $f(x)=$
• e $\mu =$
• f $\sigma =$
• g Find the probability that she is over 6.5 years.
• h Find the probability that she is between 4 and 6 years.
• i Find the 70th percentile for the age of first graders on September 1 at Garden Elementary School.