Download X P(X) Find P(X < 2) Find P(X < 2)

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7.1 Discrete and Continuous Random Variables
A random variable, X, is a numerical outcome of a
random phenomenon. For example, when a fair die is
rolled the possible values of the random variable are
X = 1, 2, 3, 4, 5, or 6.
A discrete random variable, has a countable number of
possible values. The probability distribution of X lists
the values and their probabilities. This information can
also be displayed as a probability histogram.
Example: What is the probability distribution for the
discrete random variable X that counts the number of
boys in a three-child family. Display in a histogram.
X
P(X)
Find P(X < 2)
Find P(X < 2)
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A continuous random variable X can take all the
values in a specified interval. The probability
distribution of X is described by a density curve.
Example: Does the random number generator,
rand, give a discrete or continuous random variable?
What is its distribution?
The probability of an event is given by the area
under the density curve.
Example: What is the probability of the random
number being between 0.3 and 0.7?
What is the probability of X < 0.5?
What is the probability of X = 0.5?
What is the probability of X < 0.5?
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Random variables often behave differently than
algebraic variables. For instance, in algebra,
x + x = 2x. For random variables, does X+X = 2X?
Consider the random phenomena of rolling one
dice. What is the probability model?
X
P(X)
What is the probability model for 2X?
2X
P(2X)
What is the model for X + X
X+X
P(X+X)
Find the mean and standard deviation for X, 2X,
and X+X. What do you notice about the means?
Why is there less variability when we add X+X than
for 2X?
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Normal Probability Distributions
Recall that the Normal curve we learned about in
Chapter 2 is a density curve (meaning its area = 1).
So the Normal distribution is a continuous
probability distribution. The standard Normal
random variable is given by
X 
Z

Example: Suppose that the true proportion of adults
who jog is 0.15. If we asked an SRS of 150 adults
“Do you jog?” would we expect exactly 15% to say
yes?
The actual response from the survey for repeated
samples of 150 adults will be normally distributed
with   .15 with   .029 . What is the probability
that 20% or more of a survey’s respondents say they
jog?
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Follow up: What’s the probability that between
14% and 16% of the respondents say they jog?
To simulate the jogging survey, we can use
randNorm(.15, .029)
In 100 trials, how often do you get p > 0.20?
Hint: Store the 100 trials in L1and then SortD (L1)
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The Greed Game
The goal of this game is to be the person with the
most points at the end of 5 rounds. All students
should stand up. You will start each round with 5
points. A single dice will be rolled. If the roll is a
“1” you lose all your points for that round. If the
roll is any other number, you add that number to
your score. You may sit down at any time to keep
your points. The round ends when a “1” is rolled.
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7.2 Means and Variances of Random Variables
The mean of a random variable X is the weighted
average of the possible values of X. It takes into
account that all outcomes of X are not equally likely.
Example: In Pick 3 Lottery, you choose a 3-digit
number. If your number matches the number chosen
by the lottery board, your payoff is $500. Otherwise
you win nothing. What is the mean payoff?
Payoff X:
Probability:
Mean Payoff =
If a lottery ticket costs $1, how much does the state
profit on average?
Note: In this example, The mean payoff is not a
possible outcome X.
The mean is also called the expected value, but it is
the expected value over the long-run.
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We use the symbol  X or E(x) to represent the
mean value of the random variable X.
For a discrete random variable X, with possible
outcomes x1, x2, …xk
 X  E( x)  x1 p1  x2 p2  ...xk pk
For a continuous random variable,  X lies at the
balance point of the probability distribution curve.
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
The variance of a discrete random variable is X .
 X 2   x1   X  p1   x2   X  p2  ...  xk   X  pk
2
2
2
Example: Find the standard deviation of X for the
Pick 3 game.
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The Law of Large Numbers
LoLN: As the sample size of a SRS is increased,
the sample mean x always approaches the true
mean  of the population.
Example: Suppose that the SAT Math scores of AP
Statistics students are distributed normally with a
mean of 620 and a standard deviation of 70.
Simulate drawing a sample of 100 students and
calculate their average SAT score.
Seq(X, X, 1, 100)  L1
randNorm(620, 70, 100)  L2
cumSum(L2)  L3
L3/L1  L4
Examine a scatter plot of L1 and L4 and interpret it.
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Remember:
* The Law of Large Numbers holds true regardless
of the shape of the probability distribution.
*There is no “law of small numbers” – short
sequences of random events do not follow the type
of average behavior that occur in the long run.
*LoLN is very important for inferential statistics
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Rules for Combining Means
If X is a random variable and a and b are constants
a bX  a  b X
If X and Y are random variables, then
 X Y   X  Y
Example: We know that the probability distribution
for rolling one die is
X
1
2
3
4
5
6
1
1
1
1
1
P(X) 16
6
6
6
6
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What is the mean of this distribution?  x 
Suppose we create a new random phenomenon that
doubles each roll of the die and adds 3 to it.
X
P(X)
What is the mean of this new distribution?
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Rules for Combining Variances
Rule 1: If X is a random variable and a and b are
2
2 2


b
X
a bX
constants, then
Rule 2: If X and Y are independent random
variables, then
 X2 Y   X2   Y2
 X2 Y   X2   Y2
Rule 3: If X and Y are dependent random variables
with correlation  , then
 X2 Y   X2   Y2  2 X  Y
Example: Find the mean and variance for rolling one
die and doubling the pips.
Example: Find the mean and variance for rolling two
dice and counting the pips.
Use the above example to explain why in Statistics
X  X  2X
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Why do variances add even when we are finding the
difference between two variables?
To answer this question, we are going to substitute
range as our measure of variability instead of variance.
Imagine you have a basket of grapefruit weighing
14 – 22 ounces and a basket of oranges weighing
7 – 10 ounces.
What is the range of grapefruit weights?
What is the range of orange weights?
Now suppose that you are going to randomly pick
one fruit from each basket. What is the range of the
possible sum of the two weights?
Max:
Min:
Range:
What could the range of the difference in the two
weights be?
Max:
Min:
Range: