Download D – Mean Variance sd

Discrete Random Variables
Sometimes we want to compare 2 or more frequency distributions:
Cost of living in
UK
Which is the cheapest place?
We want to refer to a measure that
Represents the whole distribution.
Arithmetic Mean
Measure of central
tendency

Expected value (E)
Arithmetic mean and Expected value
A fair dice is thrown 60 times
and the results are recorded
in a frequency distribution
table:

x
1
2
f
9
11 12 8

Mean?
3
4
5
6

NOW:
given a probability
function, calculate the
Expected value of X.
()  4
x
13 7
P( X  x)
fx


f
3.43(2dp)
Will ALWAYS be 1!

1
2
0.1 0.2
5
0.7
Multiply each value of X by its
corresponding probability
xP

Add these values

Divide by the total probability
 xP
P
Can you write a general formula for mean/expected
value for any probability distribution?
E (X )

  xP( X  x)
Find E(X) and tell which is the best

£1 gambling machine
Pay out £
x
P( X  x)
0
2
0.7 0.2
5
0.1

50 p gambling machine
Pay out £
P( X  x)
x0
1
10
0.75 0.2 0.05
E( X )  (0  0.7)  (2  0.2)  (5  0.1) E( X )  (0  0.75)  (1 0.2)  (10  0.05)
E ( X )  0.9
You will loose 10
pence on average
per game.
E ( X )  0.7
You will win 20
pence on average
per game.
Variance and Standard Deviation
Variance of a Frequency
distribution
( x   )
 
n
2
2
Standard Deviation of a
Frequency distribution
( x   ) 2

n

Find the deviation of each
possible value from the mean.
x
2
(x  )

Square each deviation

Multiply each squared deviation
by the corresponding probability
( x   ) 2  P( X  x )
Why is it unnecessary in the case of a
random variable to divide by the total
probability?
Can you write a general formula for the
variance for any probability distribution?
Var (X ) 

2
  ( x   )  P( X  x )
2
Var ( X )   x p  
2
2
Try this…


£1 gambling machine:
Find Var (X)
E ( X )  0.9
Var (X ) 
2.49
Pay out £
P( X  x)
x
0
2
0.7 0.2
5
0.1
You think you can do this…?


The probability function of the discrete
random variable X is shown in the table:
x
1
2
3
4
P( X  x)
0.2
a
0.25
b
Given E(X) = 2.95, find the values of a
and b.
A = 0.1



HINTS:
Probabilities add to 1
think about how we get E(X)
Get two equations, solve simultaneously
B = 0.45
Homework * uniform distributions =
probabilities are the same…
Exercise D page 93
Questions 3,4,5,6*,8
AND
Exam question:
A gambler plays a game in which two dice are rolled and the scores
added. If the total is 2, she loses £10. If the total is 6, she loses
£7, but if either dice show a 6, the gambler wins £6.
A. Draw up the probability distribution table of the gambler’s
losses.
B. Find the expected gain per game for the gambler
C. How much would the gambler gain if she played the game 200
times? Would you say the game is fair?