INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 4: Probability: The Study of Randomness

(Part 2)

Dr. Nahid Sultana

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Chapter 4 Probability: The Study of Randomness

4.1 Randomness

4.2 Probability Models

4.3 Random Variables

4.4 Means and Variances of Random Variables

4.5 General Probability Rules*

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4.3 Random Variables 3

Random Variable

Discrete Random Variables

Continuous Random Variables

Normal Distributions as Probability Distributions

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Random Variables

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A probability model: sample space S and probability for each outcome. A numerical variable that describes the outcomes of a chance process is called a random variable. The probability model for a random variable is its probability distribution.

The probability distribution of a random variable gives its possible values and their probabilities.

Example: Consider tossing a fair coin 3 times. Define X = the number of heads obtained.

X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH

Value 0 1 2 3 Probability 1/8 3/8 3/8 1/8

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Discrete Random Variable Two main types of random variables: discrete and continuous.

A discrete random variable X takes a fixed set of possible values with gaps between.

The probability distribution of a discrete random variable X lists the values xi and their probabilities pi:

The probabilities pi must satisfy two requirements:

1. Every probability pi is a number between 0 and 1.

2. The sum of the probabilities is 1.

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Discrete Random Variable (Cont…)

Example: Consider tossing a fair coin 3 times. Define X = the number of heads obtained.

X = 0: TTT X = 1: HTT THT TTH X = 2: HHT HTH THH X = 3: HHH Value 0 1 2 3

Probability 1/8 3/8 3/8 1/8

Q1: What is the probability of tossing at least two heads? Ans: P(X ≥ 2 ) = P(X=2) + P(X=3) = 3/8 + 1/8 = 1/2

Q2: What is the probability of tossing fewer than three heads?

Ans: P(X < 3 ) = P(X=0) +P(X=1) + P(X=2) = 1/8 + 3/8 + 3/8 = 7/8 Or P(X < 3 ) = 1 – P(X = 3) = 1 – 1/8 = 7/8

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Discrete Random Variable (Cont…)

Example: North Carolina State University posts the grade distributions for its courses online. Students in one section of English210 in the spring 2006 semester received 31% A’s, 40% B’s, 20% C’s, 4% D’s, and 5% F’s.

The student’s grade on a four-point scale (with A = 4) is a random variable X. The value of X changes when we repeatedly choose students at random , but it is always one of 0, 1, 2, 3, or 4. Here is the distribution of X:

Q1: What is the probability that the student got a B or better? Ans: P(X ≥ 3 ) = P(X=3) + P(X=4) = 0.40 + 0.31 = 0.71

Q2: Suppose that a grade of D or F in English210 will not count as satisfying a requirement for a major in linguistics. What is the probability that a randomly selected student will not satisfy this requirement?

Ans: P(X ≤ 1 ) = 1 - P( X >1) = 1 – ( P(X=2) + P(X=3) + P(X=4) ) = 1- 0.91 = 0.09

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Continuous Random Variable

A continuous random variable Y takes on all values in an interval of numbers.

Ex: Suppose we want to choose a number at random between 0 and 1. -----There is infinitely many number between 0 and 1.

How do we assign probabilities to events in an infinite sample space?

The probability distribution of Y is described by a density curve.

The probability of any event is the area under the density curve and above the values of Y that make up the event.

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A discrete random variable X has a finite number of possible values. The probability model of a discrete random variable X assigns a probability between 0 and 1 to each possible value of X.

A continuous random variable Y has infinitely many possible values. The probability of a single event (ex: X=k) is meaningless for a continuous random variable. Only intervals can have a non-zero probability; represented by the area under the density curve for that interval .

Discrete random variables commonly arise from situations that involve counting something.

Situations that involve measuring something often result in a continuous random variable.

Continuous Random Variable (Cont…)

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Continuous Probability Models

Example: This is a uniform density curve for the variable X. Find the probability that X falls between 0.3 and 0.7.

Ans: P(0.3 ≤ X ≤ 0.7) = (0.7- 0.3) * 1 = 0.4

Uniform Distribution

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Continuous Probability Models (Cont…)

Example: Find the probability of getting a random number that is less than or equal to 0.5 OR greater than 0.8.

P(X ≤ 0.5 or X > 0.8)

= P(X ≤ 0.5) + P(X > 0.8)

= 0.5 + 0.2

= 0.7

Uniform Distribution

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Continuous Probability Models (Cont…)

General Form:

The probability of the event A is the shaded area under the density curve. The total area under any density curve is 1.

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Normal Probability Model

The probability distribution of many random variables is a normal distribution.

Example: Probability distribution of Women’s height. Here, since we chose a woman randomly, her height, X, is a random variable.

To calculate probabilities with the normal distribution, we standardize the random variable (z score) and use the Table A.

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Normal Probability Model (Cont…) Reminder: standardizing N(µ,σ)

We standardize normal data by calculating z-score so that any normal curve can be transformed into the standard Normal curve N(0,1).

σµ)( −

=xz

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Normal Probability Model (Cont…)

Women’s heights are normally distributed with µ = 64.5 and σ = 2.5 in.

The z-scores for 68, And for x = 70",

4.15.2

)5.6468(=

−=z

z =(70 − 64.5)

2.5= 2.2

The area under the curve for the interval [68”,70”] is 0.9861-0.9192=0.0669.

Thus the probability that a randomly chosen woman falls into this range is 6.69%. i.e.

P(68 ≤ X ≤ 70)= 6.69%.

What is the probability, if we pick one woman at random, that her height will be between 68 and 70 inches i.e. P(68 ≤ X ≤ 70)? Here because the woman is selected at random, X is a random variable.