13.7.22!1!The Normal Distribution, the Standard Normal, the normal approximation to the Binomial!Recommended Reading in Book: Section 5.4 Stat 100 Introduction to Probability !Juana Sanchez!UCLA Dept of Statistics !Announcements!• Today: The normal distribution, the normal approximation to the binomial distribution. !• Homework 4 is due today. !• Quiz 3 is being done in the TA session today, at the beginning of session. !1. The Normal random variable!Definition. A r.v. is Normal with parameters μ and σ2 if its probability density function is given by"€ f (x) =12πσ2e−(x−µ)22σ2Height, weight, human measurements, measurement errors in scientific experiments, reaction time in psychological experiments, pollutant concentration, amounts dispensed into containers by machines, thickness of material specimens, economic measures and indicators. All share a common characteristic. They are Normal random variables.!€ −∞ < x < ∞, − ∞ <µ< ∞,σ> 013.7.22!2! ‘Bell Shaped’! Symmetrical ! Mean, Median and Mode!! are Equal!Location is determined by the mean, μ!Spread is determined by the standard deviation, σ!The random variable has an infinite theoretical range: -∞ to ∞ ! Mean = Median = Mode X f(X) μ!The Normal is a family of densities!By varying the parameters µ and σ, we obtain different normal distributions Expected Value and Variance!€ E (X ) =µVar(X ) =σ2It turns out (see proof at the end of this lesson) that !and that 68% of the values of the random variable X are within one standard deviation of μ , 95% are within 1.96 standard deviations of μ and 99.7% are within 3 standard deviations of μ. That is, most of the area under the Normal curve lies within 3 standard deviations of the mean μ. Very little probability lies past that. Also, 50% of the area lies to the right and 50% to the left of μ. Thus the Normal random variable has a symmetric pdf.!13.7.22!3!II.Linear functions of Normal r.v.’s!One very important property of the normal r.v. is that any linear function of a normal r.v., also is Normally distributed. That is, if X is normally distributed with mean μ and variance σ2 , and if Y = aX + b, then Y is also Normally distributed with E(Y) = a μ + b and Var(Y) = a2 σ 2 .!For example, if X is N(μ=2, σ2=0.2), then Y=4X -2 is N(μ=6, σ2=3.2). We use to prove this the linear property of the expectation operator and the rules of expectations we know. !Example !Suppose Y = 2X + 2 where X is N(0, 1). Y is N(μ=2, σ2 =4). The area under the X density for any two values of X will be the same as the area under the two corresponding Y points. For example, the area between -1 and 1 on the left is the same as the area between 0 and 4 on the right.!III. Finding probabilities of normal random variables!Example. Scores on an exam have mean 70 and standard deviation 6. What percentage of the class got a score higher than 76?!None of the standard integration techniques can be used to evaluate this integral. We need some computational help. This will be the Standard Normal density. !€ 12π3676∞∫⋅ e−(x−70)22⋅36dx13.7.22!4!IV. The standard Normal Density!The Standard Normal distribution is just a normal distribution with mean !0 and variance 1. It turns out that if we standardize a Normal random variable X with parameters μ, σ in the following way!The new standardized normal random variable Z follows a Standard Normal distribution with μ= 0 and standard deviation σ = 1. !€ Z =X −µσWe can prove that Z has that expected value and standard deviation by using the rules of expectations we learned earlier this quarter. !It is harder to prove that the distribution of the random variable Z is Normal. Using the Standard Normal distribution we can find probabilities for any Normal random variable.!€ E (Z) = EX −µσ⎛ ⎝ ⎞ ⎠ = E1σX⎛ ⎝ ⎞ ⎠ − Eµσ⎛ ⎝ ⎞ ⎠ =µσ−µσ= 0€ Var (Z) =X −µσ= Var1σX⎛ ⎝ ⎞ ⎠ − Varµσ⎛ ⎝ ⎞ ⎠ =σ2σ2= 1Example !If X is distributed normally with mean of 100 and!standard deviation of 50, the Z value for X =200 is!This says that X = 200 is two standard deviations (2 increments of 50 units) above the mean of 100.!€ Z =X −µσ=200 −10050= 2.013.7.22!5! Probabilities for the Standard N The Standard Normal Table ! The value within the table gives the cumulative probability up to the desired Z value!.9772 2.0 P(Z < 2.00) = 0.9772 The row shows the value of Z to the first decimal point! The column gives the value of Z to the second decimal point!2.0 . . . Z 0.00 0.01 0.02 … 0.0 0.1 Example !Example. Scores in an exam are normally distributed with μ=70 and σ=6. What is the probability that a randomly chosen student got more than 76 in the exam. !€ P(X > 76) = P z >76 − 706⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = P(z > 1) = 1− P(Z < 1)= 1− 0.84 = 0.16Finding the Probabilities under the Standard Normal requires that we use tables or some calculator or computer. In this class, during exams, we will use the table posted. Let’s go now to the Appendix. !General Procedure for Finding Probabilities for Normal X! Draw the normal curve for the problem in terms of X! Translate X-values to Z-values! Use the Standardized Normal Table!To find P(a < X < b) when X is distributed normally:13.7.22!6!One more example. The time it takes a driver to react to the break light on a decelerated vehicle is critical in avoiding rear-end collisions. Someone suggests that reaction time for an in-traffic response to a brake signal from standard brake lights can be modeled with a normal distribution having parameters μ = 1.25 sec and σ = 0.46 sec. In the long run, what proportion of reaction times will be between 1.00 sec and 1.75 sec? Let X denote reaction time. !€ P(1 < x < 1.75) = P(−0.54 < Z < 1.086) = 0.5675Finding Normal Probabilities!Suppose X is normal with mean 8.0 and standard deviation 5.0 Find P(X < 8.6)!X!8.6!8.0!Suppose X is normal with mean 8.0 and standard deviation 5.0. Find P(X < 8.6)!Z!0.12! 0!X!8.6! 8!Finding Normal Probabilities!P(X < 8.6)! P(Z < 0.12)!€ Z =X −µσ=8.6 − 8.05.0= 0.1213.7.22!7!Z!0.12!Z! .00! .01!0.0!.5000!.5040! .5080!.5398!.5438!0.2! .5793! .5832! .5871!0.3! .6179! .6217! .6255!Solution: Finding P(Z < 0.12)!.5478!.02!0.1!.5478!Standardized Normal Probability !Table (Portion)!0.00!= P(Z < 0.12)!P(X <
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