PROBABILITY AND STATISTICS

WEEK 4

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Random Variable

Random Variable. Let S be the sample space for an experiment. A real-valued function that is defined on S is called a random variable.

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Random Variable

1.Discrete Random Variable: Has a finite (or countably infinite) range.•Tossing a coin: X= 0 for head and X= 1 for tail

2.Continuous Random Variable: Has an interval of real numbers for its infinite range.•The life length of a light bulb: X ≥ 0

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Reminder !

2. s2 and s are the variance and standard deviation of the sample

x3. , s2, and s are called sample statistics

4. (lowercase Greek letter “mu”) is the mean of the population

5. 2 (“sigma squared”) is the variance of the population

6. (lowercase Greek letter “sigma”) is the standard deviation of the population

7. , 2, and are called population parameters. (A parameter is a constant. , 2, and are typically unknown values.)

x1. is the mean of the sample

Discrete Random Variables

Let 4 coins tossed, and let X be the number of heads that are obtained. Let us find the distributions of that experiment.

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Probability Distribution

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Discrete Random Variables

ExampleThree balls, a, b, c, are randomly distributed in three boxes. Determine the distribution of the random variable X ="the number of non-empty boxes".

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Discrete Random Variables

ExampleConsider a group of five potential blood donors; “a, b, c, d, and e” of whom only a and b have type 0+

blood. Five blood samples, one from each individual, will be

typed in random order until an 0+ individual is identified. Let the rv Y=“the number of typings necessary to identify an 0+ individual.”

Find the pmf.

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The Cumulative Distribution Function

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Example

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The Expected Value of X(Mean of a Discrete Random Variable)

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[ ( )]x.p x

The Expected Value of X(Mean of a Discrete Random Variable)

• The mean, , of a discrete random variable x is found by multiplying each possible value of x by its own probability and then adding all the products together:

Notes: The mean is the average value of the random variable, what happens on average

The mean is not necessarily a value of the random variable

The Variance of X

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Example

The grades of n = 50 students in a statistics class are summarized as follows:

Find the pmf, mean, variance and sd.

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Grade (X)

1 2 3 4

Number of Students

10 20 15 5

Example

Variance and Standard Deviation of a Discrete Distribution. Suppose that a random variable X can take each of the five values −2, 0, 1, 3, and 4 with equal probability.

Determine the variance and standard deviation of X.

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A Shortcut Formula for V(X)

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Example

Determine the mean, variance, and standard deviation of casting a single die (X).

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Example

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Example

A shipment of 8 similar microcomputers to contains 3 defective one. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives.

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Example

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P xx

x( ) 815

for 3, 4, 5, 6, 7

ExampleExample: The probability distribution for a random

variable x is given by the probability function:

Find the mean, variance, and standard deviation

Discrete Uniform Distribution

A discrete uniform random variable X has an equal probability for each value in the range of X= [a, b], a < b. Thus, the probability mass function of X is;

P(x)= 1/(b-a+1) where x=a,a+1,…,b

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Example

• Casting a die…

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Example

Suppose that product codes of 2, 3, or 4 letters are equally likely.

• Determine the probability mass function of the number of letters (X) in a product code.

• Calculate the mean and variance of X

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