Post on 18-Jan-2016
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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.
Onur Doğan
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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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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[ ( )]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
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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Example
Determine the mean, variance, and standard deviation of casting a single die (X).
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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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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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