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lecture 10

Lecture 9

The probability distribution for The probability distribution for the discrete variablethe discrete variable

Satify the following conditions

P(x)>= 0 for all x

Probability Functions and Probability Distributions

Example 1:

In an experiment of tossing a fair coin three times observing the number of heads (X) find:

1. The probability distribution table

2. The mathematical expectation ( mean)

3. Construct the probability histogram

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S = { HHH , HHT , HTH , THH , TTT , THT , TTH , HTT }

At any point x, the number of At any point x, the number of heads areheads are

S = { HHH , HHT , HTH , THH , TTT , THT , TTH , HTT }

23 022 11 1

From the sample space

1. The probability distribution table

33 22 11 00 XXقيمة قيمة

1/8 3/8 3/8 1/8 P(X=X)

3 2 1 0 عدد ظهور Hالوجه

1/8 3/8 3/8 1/8 االحتمال

Mathematical

Expectation

2. The mathematical expectation ( mean)

ProbabilProbability ity

HistograHistogramm

3. probability histogram

.

Example 2:

In an experiment of tossing a fair coin three times observing the absolute difference between the number of H and T find:

1. The probability distribution table

2. The expectation

3. The mean

4. Construct the probability histogram

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S = { HHH , HHT , HTH , THH , TTT , THT , TTH , HTT }

13 311 11 1

Its range is a random variable defined in Y= {1,3}

The absolute difference and the range

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3 1 YYقيمة قيمة

1/4 3/4

P(Y=y)

1. The probability distribution table

What is the probability that Y= 1

Mathematical Expectationمثال

المجموع 3 1 YYقيمة قيمة

1 1/4 3/4 P(y)

6/4 3/4 3/4 Y p(y)

In an experiment of rolling two fair dice, X is defined as the sum of two up faces

مثال

11 22 33 44 55 66

11 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

22 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)

33 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)

44 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4)

55 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5)

66 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

Elements of the sample space = 62 = 36 elements

X is a random variable defined in SThe range of it is {2,3,4,……….,11,12}

What is the probability that X= 4i.e what is the probability that the sum of the two upper faces =4