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BUAD 300Prof. R. RobinsonHandout on Normal Probability DistributionsStudents in this course have successfully completed an elementary 200-level statistics course. They should recall the elements of applying the normal distribution to statistical problems as covered in Chapter 6 of the text for this course. This material is briefly reviewed here for the purpose of reinforcement. The Normal Distribution A random variable is defined as one that takes on numerical values according to some probability distribution, which means that the values of the random variable depends upon the outcome of some chance event such as the roll of a die, the future state of the economy, or other uncertain event. Probabilities can be assigned to these outcomes.The probability distribution shown in Graph 1 is the “normal” distribution, a symmetric distribution frequently encountered in the natural and social world. In this graphic, values for the variable X are measured on the horizontal axis, and the probabilities of obtaining these values are measured on the vertical axis. The parameter termed the “expected value” is a measure of central tendency and is symbolized by E(X). This distribution, like many others, is a symmetric distribution in that the left half of the distribution is a mirror image of the right half. For symmetric distributions, the expected value is in the center. Also, for the normal distribution, the expected value has the highest probability associated with it.Graph 1The Normal Distribution927735149860E(X)Probability of X00E(X)Probability of X619125109220Prob ≈ .34500Prob ≈ .3452501265111125001390650254000022574258445500295275151130Inflection point00Inflection point208597586360009906002921000219075101600Prob ≈ .16Prob ≈ .161781175254000198120050165008858251778019526256032500534733548895X00X E(X) - σFor symmetric distributions, if one marks off equal distances above and below the expected value, for example the values E(X) + c, and E(X) – c where c is just some fixed number, then the probability in the tails are equal. This stems from the “mirror image” nature of symmetric distributions referred to above. The variance of a probability distribution is its “measure of dispersion about the mean.” The variance is a measure of how wide the distribution is spread. it is usually symbolized by the square of the Greek letter sigma, 2, or by Var(X). The standard deviation is the square root of the variance which is symbolized by . For the normal distribution, is the distance between the expected value and the inflection point, the point where the curvature of the probability function changes. For the normal distribution, the larger the variance the smaller the probability associated with the expected value.Probability distributions can be categorized into two types: discrete distributions and continuous distributions. Variables that behave according to the former can only take on certain “discrete” values such as 1, 2, 3, … Such a distribution, for example, could be generated by the throwing of a die. The continuous distribution is characterized by the variable being capable of taking on any value over the range of the distribution, no matter how many decimal places are included in the value. The normal distribution is a continuous distribution. In either the continuous or discreet case, the sum of probabilities for all possible outcomes must be one. For the continuous case, this means that the entire area under the distribution curve (as illustrated by Graph 1), usually termed the density function, must equal one.The normal density function is given in your text on page 272 of the12th edition. This expresses the function that generates the normal curve. Section 6.2 of your text explains the normal distribution. This density function for the normally distributed variable x with an E(x) = μ and a standard deviation of σ is given below as f(x). Note that E(x) and μ are interchangeable in the sense that they are identical metrics.f(x) = 1σ√2π e-z2/2 where z = x- μσ, E(x) = μ, “e” is the natural-log base, and π is the familiar 3.14159….We must use integral calculus in order to calculate the probability of x occurring over some range, although this integration is cumbersome. Using integral calculus, if f(x) gives the normal density probability, then for a x b, Prob (a x b) = abfxdx. (Note: “Prob (a ≤ x ≤ b” should be read as “the probability of x being in the range a ≤ x ≤ b.”) Fortunately, any normal distribution can be changed, via the z equation, into the standard normal distribution, which is easy to use for probability calculations. Note that z is termed standard normal metric, which you are familiar with from your 200-level statistics course. You used this “z” to find normal probabilities for various ranges of values for x as found in the standard normal tables. (See the inside cover of your text.)Review questions:Suppose a normal distribution has an expected value of D. If the probability of obtaining a value greater than D+d is , what is the probability of obtaining a value less than D-d? What is the probability of obtaining a value between D and D+d? Draw the distribution and properly label it.If f(x) is the normal density, (a) what is f(x) at x = μ and σ = 1? (b) What is f(x) at z = 1 and σ = 1?Since at x = E(x), then z = x- E(x)σ = 02 = 0, and at σ = 1, we have f(x) = 1σ√2π e-z2/2 = 1√2π e0 = 1√2πAt z = 1 and σ = 1, them f(x) = 1σ√2π e-z2/2 = 1√2π e-1If x is normally distributed, and E(x) = 10, and σx = 2 then what is Prob(8 ≤ x ≤ 12)? Note that at x = 8, then z = x- E(x)σ = 8- 102 = -1, and at x = 12, z = x- E(x)σ = 12 - 102 = +1. Find the answer for Prob(8 ≤ x ≤ 12) from the inside cover of your text, and also from Graph 1.Using the standard normal distribution:Any normal distribution for which we know the expected value and standard deviation can be transformed into a standard normal distribution, or z statistic. In this z statistic given below, x is normally distributed, and Hyp. x is some hypothetical measure for which we want to calculate the probability of obtaining some measure that is either less than, or greater than this Hyp. x.z = Hyp.X - E(X)σX Review Question: Note that to answer questions concerning the normal distribution, it is best to start by drawing the distribution, and properly labeling it.X is normally distributed with E(X) = 10 and σX = 2. What is Prob(X ≥ 12)? z = Hyp.X - E(X)σX = 12-102 = 1.We find from the standard normal tables that Prob(X ≥ 12) ? .16. For a value of Hyp. x equal to E(x), i.e. Hyp. x = E(x), what is f(x) if σ = 2? Note: Is this similar to question 2a presented above.Suppose X is normally distributed with an expected value of D and a standard deviation of σ = 1. What is prob(X ≤ D – σ)? What is prob(D – σ ≤ X ≤ D)?Homework questions due as assigned in class. Use pencil and eraser only, and be very neat. Write the appropriate equation first, then continue the equation by incorporating the numerical data, and then solve as illustrated by the solution presented above to #3. If x is normally distributed, and Hyp. X = 12, E(X) = 8, σx = 2, what is prob(x ≤ Hyp. X)? What is prob (x > Hyp. X)? (Find the answer from the z tables on the inside cover of your text.)A random variable Y is normally distributed with E(Y) = 50, and σY = 5. What is the probability of Y < 40? What is the probability that Y ≥ 40? What is the probability that 40 ≤ Y < 50?A random variable Y is normally distributed with E(Y) = 80, and σY = 10. What is the probability of Y < 58? What is the probability that Y ≥ 58? What is the probability that 58 ≤ Y < 80?(Note that students in this class need not know the following proof, although it should be of interest to those knowledgeable of calculus.) Proof that the inflection points for the normal density function is at μ ±σ:Allow the normal density to be given by f(x) as specified below:f(x) = 1σ√2π e-z2/2 where z = x- μσDifferentiating with respect to x gives us df/dx where df/dx = -z(dz/dx)f(x) = - z f(x)σand d2f/dx2 = f(x)σ2 [z2 – 1] For the inflection point, it must be that d2f/dx2 = 0, so that d2f/dx2 = 0 iff z2 = +1 which requires that z = ±1.Since z = ±1 occurs if (x - μ)2 = σ2, then x = μ±σ. ................
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