1. 4.1B – Probability Distribution
MEAN of discrete random variable:
µ = ΣxP(x)
EACH x is multiplied by its probability and the products are added.
µ = EXPECTED VALUE of discrete random variables

2. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24 2 3342 4 30 5 21 Σ=

3. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24 2 3342 4 30 5 21
Σ=150

4. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16 2 33 3 42 4 30 5 21
Σ=150

5. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16 2 33
.22 3 42
.28 4 30 .2 5 21
.14
Σ=150

6. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16 2 33
.22 3 42
.28 4 30 .2 5 21
.14
Σ=150
Σ=1.0

7. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22 3 42
.28 4 30 .2 5 21
.14
Σ=150
Σ=1.0

8. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28 4 30 .2 5 21
.14
Σ=150
Σ=1.0

9. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28
3(.28)=.84 4 30 .2 5 21
.14
Σ=150
Σ=1.0

10. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28
3(.28)=.84 4 30 .2
4(.2)=.80 5 21
.14
Σ=150
Σ=1.0

11. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28
3(.28)=.84 4 30 .2
4(.2)=.80 5 21
.14
5(.14)=.70
Σ=150
Σ=1.0

12. Example: Find the Mean Score, x Frequency, f P(x) f/Σf xP(x) 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28
3(.28)=.84 4 30 .2
4(.2)=.80 5 21
.14
5(.14)=.70
Σ=150
Σ=1.0
ΣxP(x)=2.94
= µ

13. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
$248
$148
$73
$-2
Prize-$2
EV = µ=ΣxP(x)

14. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
$248
$148
$73
$-2
Prize-$2
EV = µ=ΣxP(x)

15. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
$248
$148
$73
$-2
Prize-$2
EV = µ=ΣxP(x)

16. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
$248
$148
$73
$-2
Prize-$2
EV = µ=ΣxP(x)

17. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
$248
$148
$73
$-2
Prize-$2
EV = µ=ΣxP(x)

18. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
$248
$148
$73
$-2
1496/1500
Prize-$2
EV = µ=ΣxP(x)

19. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
$148
$73
$-2
1496/1500
Prize-$2
EV = µ=ΣxP(x)

20. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
248(1/1500)=248/1500
$148
$73
$-2
1496/1500
Prize-$2
EV = µ=ΣxP(x)

21. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
248(1/1500)=248/1500
$148
148(1/1500)=148/1500
$73
$-2
1496/1500
Prize-$2
EV = µ=ΣxP(x)

22. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
248(1/1500)=248/1500
$148
148(1/1500)=148/1500
$73
73(1/1500)=73/1500
$-2
1496/1500
Prize-$2
EV = µ=ΣxP(x)

23. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
248(1/1500)=248/1500
$148
148(1/1500)=148/1500
$73
73(1/1500)=73/1500
$-2
1496/1500
-2(1/1500)=-2992/1500
Prize-$2
EV = µ=ΣxP(x)

24. Example: Find the EXPECTED VALUE of your gain.
1500 tickets are sold for $2 each for 4 prizes of $500, $250, $150, and $75. You buy 1 ticket. What is the expected value of your gain?
Gain, x
P(x)
xP(x)
$498
1/1500
498(1/1500)=498/1500
$248
248(1/1500)=248/1500
$148
148(1/1500)=148/1500
$73
73(1/1500)=73/1500
$-2
1496/1500
-2(1496/1500)=-2992/1500
Prize-$2
EV = µ=ΣxP(x)=-2025/1500
= -$1.35

25. Standard Deviation VARIANCE of discrete random variable
σ² = Σ(x-µ)²P(x) OR
σ² = [Σx²P(x)] - µ²
STANDARD DEVIATION of discrete random variable
σ = √σ²

26. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16 2 33
.22
2(.22)=.44 3 42
.28
.84 4 30
.20
.80 5 21
.14
.70
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

27. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94 2 33
.22
2(.22)=.44 3 42
.28
.84 4 30
.20
.80 5 21
.14
.70
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

28. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94 2 33
.22
2(.22)=.44
2-2.94=
-.94 3 42
.28
.84 4 30
.20
.80 5 21
.14
.70
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

29. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94 2 33
.22
2(.22)=.44
2-2.94=
-.94 3 42
.28
.84
.06 4 30
.20
.80 5 21
.14
.70
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

30. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94 2 33
.22
2(.22)=.44
2-2.94=
-.94 3 42
.28
.84
.06 4 30
.20
.80
1.06 5 21
.14
.70
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

31. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94 2 33
.22
2(.22)=.44
2-2.94=
-.94 3 42
.28
.84
.06 4 30
.20
.80
1.06 5 21
.14
.70
2.06
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

32. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764 2 33
.22
2(.22)=.44
2-2.94=
-.94 3 42
.28
.84
.06 4 30
.20
.80
1.06 5 21
.14
.70
2.06
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

33. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884 3 42
.28
.84
.06 4 30
.20
.80
1.06 5 21
.14
.70
2.06
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

34. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884 3 42
.28
.84
.06
.004 4 30
.20
.80
1.06 5 21
.14
.70
2.06
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

35. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884 3 42
.28
.84
.06
.004 4 30
.20
.80
1.06
1.124 5 21
.14
.70
2.06
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

36. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884 3 42
.28
.84
.06
.004 4 30
.20
.80
1.06
1.124 5 21
.14
.70
2.06
4.244
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

37. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884 3 42
.28
.84
.06
.004 4 30
.20
.80
1.06
1.124 5 21
.14
.70
2.06
4.244
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

38. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004 4 30
.20
.80
1.06
1.124 5 21
.14
.70
2.06
4.244
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

39. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004
.001 4 30
.20
.80
1.06
1.124 5 21
.14
.70
2.06
4.244
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

40. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004
.001 4 30
.20
.80
1.06
1.124
.225 5 21
.14
.70
2.06
4.244
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

41. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004
.001 4 30
.20
.80
1.06
1.124
.225 5 21
.14
.70
2.06
4.244
.594
Σ=150
Σ=1.0
Σ=2.94 =µ
σ = √σ²

42. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004
.001 4 30
.20
.80
1.06
1.124
.225 5 21
.14
.70
2.06
4.244
.594
Σ=150
Σ=1.0
Σ=2.94 =µ
Σ=1.616
σ = √σ²

43. Example: Find Variance & Standard Deviation
Score, x
Freq. f
P(x)
f/Σf
xP(x)
x-µ
x-ΣxP(x)
(x-µ)²
P(x)(x-µ)² 1 24
24/150=.16
1(.16)=.16
1-2.94=
-1.94
(-1.94)²=
3.764
.16(3.764)=
.602 2 33
.22
2(.22)=.44
2-2.94=
-.94
(-.94)²=
.884
.22(.884)=
.194 3 42
.28
.84
.06
.004
.001 4 30
.20
.80
1.06
1.124
.225 5 21
.14
.70
2.06
4.244
.594
Σ=150
Σ=1.0
Σ=2.94 =µ
Σ=1.616
σ = √σ²
= √1.616 = 1.27