Deck 11: Sums of Independent Random Variables and Limit Theorems

Question 1

At a certain Doggy Weight Loss clinic, the dogs' weights are normally distributed with mean 81 pounds and standard deviation 12 pounds. If you randomly select 12 dogs, what is the probability that the average weight of those 12 dogs is at most 80 pounds.

Answer 1

.1587.

Question 2

X is a random variable with moment generating function $M _ { X } ( t ) = \left( \frac { 1 } { 4 } + \frac { 1 } { 4 } e ^ { t } + \frac { 1 } { 2 } e ^ { 2 } t \right) ^ { 8 }$ Find Var(3X) and P(X≤2).

Answer 2

Var(3X)=33/2

Question 3

The moment generating function of a random variable is $M _ { X } ( t ) = \frac { 1 } { 8 } e ^ { t } + \frac { 1 } { 2 } e ^ { 4 t } + \frac { 1 } { 4 } e ^ { 5 t } + \frac { 1 } { 8 } e ^ { 7 t }$ . Find P(X=i) for 1=1,…,7.

Answer 3

P(X=1)=1/8, P(X=4)=1/2, P(X=5)=1/4, P(X=7)=1/8, P(X=2)=P(X=3)=P(X=6)=1/8.

Question 4

The arm span of kindergarteners (in inches) is modelled by the random variable X, where X is N(2,.5). Find the probability that 10 kindergarteners can encircle a 21-feet tree in circumference by touching fingertips in a circle.

Question 5

Mary drives her car every day. The error made by her car's odometer on a random day has standard deviation 2.3 miles. Using Chebyshev's inequality, find an upper bound for the probability that after 20 days, her odometer is off by at least 15 miles.

Question 6

For a given fishing tournament, the fish caught have lengths distributed with mean 2.3 feet and standard deviation .4 feet. If you want to be 90% sure of catching the longest fish at the tournament, what should the minimum length of the longest fish you catch be?

Question 7

For a random variable Y, the moment generating function is My(t)=exp(2t²+3t). Find E(Y) and σy.

Question 8

Let the moment generating function of a random variable X be given by $M _ { X } ( t ) = \left( \frac { 3 } { 4 } e ^ { t } + \frac { 1 } { 4 } \right) ^ { 9 }$ . Find P(X>7).

Question 9

The moment generating function for a random variable is given by $M _ { X } ( t ) = \exp \left( 3 e ^ { t } - 3 \right)$ . Find P(X>2).

Question 10

At a certain drag race, finishing times (in seconds) for Japanese cars and American cars are N(15,2) and N(16.1,3), respectively. If 9 American cars and 9 Japanese cars race, find the probability that the average race time of the Japanese cars is at least 2 seconds faster than that of the American cars.

Question 11

Suppose that all we know about the heights of trees in a redwood forest is that their average height is 162 feet with standard deviation 22. Let H be the average height of 55 trees chosen at random. Estimate P(160

Question 12

Suppose that, on a random day in January, the average temperature in Duluth, MN is 0∘ F with standard deviation 4∘. Give an upper bound for the probability that a random day in January in Duluth, MN is at least 12∘.

Question 13

Let X be a random variable with moment generating function $M _ { X } ( t ) = \frac { 4 } { \left( 2 - \frac { t ^ { 2 } } { 3 } \right) ^ { 2 } }$ for t<2. Find the moments of X.

Question 14

For a gambling game, a person wants to estimate the percentage of the times a dart player hits the bullseye. How many darts should the player throw to be 96% sure that the estimate is within .02 of the actual percentage?

Question 15

Suppose that at a certain school, all 1500 students write letters to pen pals. The number of letters each student sends is independent of the number of letters other students send. Furthermore, the numbers of letters written by students are identically distributed random variables with mean 8 and standard deviation 2.2. Find the 90th percentile of the numbers of letters written.