Chapter 7.2 A: Means and Variances of Random Variables
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1. Explain the difference between the notations 𝑥̅ and 𝜇𝑥 .
𝑥̅ stands for the mean of a sample and 𝜇𝑥 refers to the mean of a population.
2. What is meant by the expected value of X ? The expected value is what you would expect to happen on average over
the long run.
3. How do you calculate the mean of a discrete random variable X ? You would take each event, multiply it by the
probabilities of them happening, and then sum these answers together.
4. Explain how to calculate the variance of a discrete random variable X using the formula. 𝜎𝑥2 = ∑(𝑥𝑖 − 𝜇𝑥 )2 𝑝𝑖 This
formula is saying that you will take an event and subtract it from the mean to get the distance of that variable from the mean.
Then we square that distance to get rid of the negative, and multiply that answer by the probability of that event occurring. You
do this for each event, and then sum those answers together to get the variance.
5. Given the variance of a random variable, explain how to calculate the standard deviation. If you have the variance,
just take the square root of the variance to determine the standard deviation.
6. Explain the Law of Large Numbers. The law of large numbers explains what we expect to see happen over the long
run. If we were to simulate an event a large number of times, it will start to follow a predicted distribution. For example, if we
flip a coin many times, we would expect to get heads half the time and tails half the time. This is not true for a small number of
trials.
7. Find the expected number of girls and the standard deviation of the random variable for #6 in homework 7.1 A.
8. In an AP Stats class, the students had the following grade distribution: (A = 4, B = 3, and so on)
Grade
0
1
2
3
4
Probability
0.01
0.05
0.30
0.43
0.21
Find the average (that is, the mean or expected grade) and the standard deviation for grades in this course.
9. Refer back to the distributions of the number of rooms for owner-occupied units and renter-occupied units in San
Jose, California from problem 8 in 7.1 A.
a)
Find the mean number of rooms for both types of housing units. How do the means reflect the differences
between the distributions you found in problem 8 from 7.1 A?
b)
Which of the two distributions for room counts appears more spread out in the probability histograms? Find the
standard deviation for both distributions.
10. The Tri-State Lottery games offers a choice of several bets. You choose a three-digit number, at the end of each
day. The lottery commission announces the winning three-digit number, chosen at random, at the end of the day. The
“box” pays $83.33 if the number you choose has the same digits as the winning number, in any order. Find the
expected payoff for a $1 bet on the box. (Assume that you choose a number having three different digits.)
11. a) A gambler knows that red and black are equally likely to occur on each spin of a roulette wheel. He observes
five consecutive reds and bets heavily on red at the next spin. Asked why, he says that “red is hot” and that the run of
reds is likely to continue. Explain to the gambler what is wrong with this reasoning.
b) After hearing you explain why read and black remain equally probable after five reds on the roulette wheel, the
gambler moves to a poker game. He is dealt five straight red cards. He remembers what you said and assumes the next
card dealt in the same hand is equally likely to be red or black. Is the gambler right or wrong? Why?
12. One consequence of the law of large numbers is that once we have a probability distribution for a random variable,
we can find its mean by simulating many outcomes and averaging them. The law of large numbers says that if we take
enough outcomes, their average value is sure to approach the mean of the distribution.
I have a bet to offer you. Toss a coin 10 times. If there is no run of three or more straight heads or tails in the 10
outcomes, I’ll pay you $2. If there is a run of three or more, you pay me just $1. Surely you will want to take
advantage of this game?
Simulate enough plays of this game (the outcomes are +$2 if you win and -$1 if you lose) to estimate the mean
outcome. Is it to your advantage to play? (You may use the binomial distribution instead of doing a simulation if you
prefer)