a a x b otherwise E X a b 2 V ar X b a 2 12Example The flight time between two

# A a x b otherwise e x a b 2 v ar x b a 2 12example

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a a x b 0 otherwise E ( X ) = a + b 2 V ar ( X ) = ( b a ) 2 12 Example The flight time between two cities is known to be uniformly distributed between 60 and 75 minutes. X: fight time (in minutes) (a) f ( x ) (b) P (69 < X < 73) (c) P (69 < X 73) (d) P ( X = 70) (e) P (69 < X < 80) (f) P (55 < X < 75) 5
Def : If N independent replications of the experiment are carried out, then the expected value (mean) μ = E ( X ), of the rendom variable X is the average of the values taken as the number of replications becomes infinitely large. μ = E ( X ) = −∞ xf ( x ) dx σ 2 = E ( X μ ) 2 = −∞ ( x μ ) 2 f ( x ) dx 6
Section 5.3 The Normal distribution A Normal probability distribution function, when plotted, gives a bell-shaped curve such that 1. The total area under the curve is 1. 2. The curve is symmetric about the mean. 3. The two tails of the curve extend indefinitely. f ( x ) = 1 2 πσ exp 1 2 σ 2 ( x μ ) 2 where μ = mean = E ( X ), σ 2 = V ar ( X ), σ = standard deviation X N ( μ, σ 2 ) 7
The most important probability distribution for describing a continuous random variable is the normal probability distribution. The normal distribution has been used in a wide variety of practical applications in which the random variables are heights and weights of people, test scores, scientific measurements, amounts of rainfall, and other similar values. 8
Def. The normal distribution with mean μ = 0 and standard deviation σ = 1 is called the standard normal distribution . That is, Z follows a standard normal distribution, Z N (0 , 1). X N ( μ, σ 2 ): f ( x ) = 1 2 πσ exp 1 2 σ 2 ( x μ ) 2 Z N (0 , 1): f ( z ) = 1 2 π exp 1 2 z 2 p. 718: Table 1: Cummulative Distribution Function, F ( z ), of the Standard Normal Distribution Table The entreis in the table give the area under the standard normal curve from −∞ to z , that is, F ( z ) = P ( Z < z ).

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