A
coin is tossed 4 times. Let X be the number
of heads. The table below shows the
probability distribution for X.
Number of
Heads
0
1
2
3
4
Probability
.0625
.25
.375
.25
.0625
 A.
Does this represent a legitimate
probability distribution? Why or why not?
 B. Find P(X ≥ 2)
 C. Find P(X ≥ 1)
 D. Find P(X ≠ 3)
1. A) 1/3
B-C) Individual answers
2. Done in class
3. A) 1%
B) sum = 1
C) 0.94
D) 0.86 E) 0.06
4. A) 0.4
B) 0.6
C) 0.2
D) 0.1
E) 0.487 F) 0.49
G) 0.73
H) 0.73
I) 0.2
J) 0.5
K) 0
5. B) 1/36
D) 2/9
E) 5/6
6. Done in class
7. A) .752
B) yes, sum = 1
C) 0.017
D) 0.024
E) 0.931
 We
use the Greek letter μ (mu)
 More specifically we will be talking about the
mean of X, so we use μx.
 The
mean of a random variable is also called
the expected value of X.
 Suppose
that X is a discrete random variable
whose distribution is
Value of X
x1
x2
x3
…
xi
Probability
p1
p2
p3
…
pi
 To
find the mean of X, multiply each possible
value by its probability, then add all the
products:
 X  x1 p1  x2 p2  ...  xk pk
  xi pi
 Recall
the example we looked at yesterday of
the distribution of the size of households
according to Census Bureau studies:
Inhabitants
1
2
3
4
5
6
7
Probability
.25
.32
.17
.15
.07
.03
.01
 Find
the mean size of an American
household.
 Draw
independent observations at random
from any population with finite mean μ.
Decide how accurately you would like to
estimate μ. As the number of observations
drawn increases, the mean x of the
observed values eventually approaches the
mean μ of the population as closely as you
specified and then stays that close.

Suppose X is a discrete random variable whose
distribution is
Value of X
x1
x2
x3
…
xi
Probability
p1
p2
p3
…
pi

And that μ is the mean of X. The variance of X is:
 x2  x1   X  p1  x2   X  p2  ...  xk   X  pk
2
2
  xi   X  pi
2
2

The standard deviation of σX of X is the square
root of the variance.
 In
1964, New Hampshire started the lottery
fad. In the Tri-State Pick 3 game that New
Hampshire shares with Maine and Vermont
the participant picks a three-digit number;
the state chooses a three-digit winning
number at random and pays you $500 if the
number is chosen. Let X be the amount of
money you win. Show the probability
distribution of X.
 Is this a legitimate probability distribution?
 Find the mean μX
 Find the standard deviation σX
 Natalie
sells Bean Bag Chairs at the local
Pack ‘n Swap. The table below shows the
distribution of the number of bean bag chairs
sold per day.
Number of
Bean Bag
Chairs Sold
5
6
7
8
9
10
Probability
.05
.15
.20
.30
.20
.10
 Is
this a legitimate probability distribution?
 Find the mean μX for the number of bean bag
chairs sold per day.
 Find the standard deviation.