Stat250.3
October 1, 2003
Activity 4: Random Variables
DISCRETE RANDOM VARIABLES
Part 1: Are these situations Binomial Experiments? If so, identify the Binomial Random Variable.
You have a die. Your goal is to roll the die three times and determine if a six appears on each roll. To do this,
you roll the die and observe if the number is a six and repeat this procedure two more times.
Is n fixed? Are there two outcomes? Are the trials independent?
Does the probability of a success remain the same for each trial?
Is this a Binomial Experiment? If so, what is the Binomial Random Variable?
You have a deck of 52 cards. The goal is to pick six cards and determine if a spade appears on each pick. You
shuffle the deck of cards, pick the top card and examine this top card to determine if it is a spade. You then
put the card aside and reshuffle the deck. You repeat this process five more times.
Is n fixed? Are there two outcomes? Are the trials independent?
Does the probability of a success remain the same for each trial?
Is this a Binomial Experiment? If so, what is the Binomial Random Variable?
Part 2: For each of the following situations, assume X is a binomial random variable with n=15 and
p=.7. Find the requested probabilities. Note that using the binomial tables you get probabilities of the
form P(X≤k).
1. P(X=10) =
2. P(X<10 ) =
3. P(4 ≤ X ≤ 12) =
4. P(4 < X ≤ 12) =
5. P(X > 2) = 1-P(X≤2) =
Stat250.3
Part 3:
A) Let X = the sum of a roll of 2 six sided die.
Is X binomial? Why or why not?
B) What is the PDF of X?
k
P(X=k)
C) What is the CDF of X?
k
P(Xk)
D) What is the probability of getting an 8?
E) What is the probability of getting 4 or less?
F) What is the expected value of X?
October 1, 2003
Stat250.3
October 1, 2003
Part 4:
A) Let X = the number of people who are registered Republicans in a random sample of 5 people. Is
X binomial why or why not?
B) Suppose the probability that a randomly selected individual is Republican is 0.2. (That is 20% of
all Americans are registered Republicans). What is the PDF of X?
k
P(X=k)
C) What is the CDF of X?
k
P(X≤k)
D) What is the probability that X = 3?
E) What is the probability that the sample proportion of Republicans is greater than 40%?
F) What is the expected value of X? The standard deviation of X?
Stat250.3
October 1, 2003
CONTINUOUS RANDOM VARIABLES:
Part 5:
Suppose that the IQ scores of PSU students are normally distributed with mean 100 and standard
deviation 12 (this is completely fictitious). NOTE: If you cannot visualize the situation, draw the
picture!!!
a. What is the z-score for Joe, who has an IQ of 122?
b. What is the probability that we will randomly select an individual with an IQ lower than Joe?
c. Susan has an IQ of 92. Find the proportion of PSU students that has an IQ score between those
of Susan and Joe.
d. What proportion of PSU students has an IQ higher than that of Susan?
e. What is the 65th percentile for the IQ scores of PSU students? (i.e. what is the IQ score that 65%
of the students have IQ score lower than that?)
Note: Here we need to work “backwards”. Now we know the probability and we want to find the
value to be less than…Instead of starting with P(X<”value”) = P% where P% is some unknown
percent we start with P% known and try to find this “value”.
f. What is the percentile ranking for an IQ score of 133?