5 Probability densities
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1 ENGG450 robability and Statistics or Engineers Introduction 3 robability 4 robability distributions 5 robability Densities Organization and description o data 6 Sampling distributions 7 Inerences concerning a mean 8 Comparing two treatments 9 Inerences concerning variances A Random rocesses
2 5 robability densities 5. Continuous random variables 5. The normal distribution 5.3 The normal approimation to the binomial distribution 5.5 The uniorm distribution 5.0 Joint distribution discrete and continuous Introduction 3 robability 4 robability distributions 5 robability densities Organization & description 6 Sampling distributions 7 Inerences.. mean 8 Comparing treatments 9 Inerences.. variances A Random processes
3 (4. Random variables (3 A random variable is deined by the ollowing:. An eperiment that can be repeated. The set that contains all possible outcomes o the eperiment is called the sample space. A subset o the sample space is called an event. The outcome o an eperiment is a number which is a sample o the random variable.. A unction that associates a probability to each event. Eperiment obtain a sample o the random variable outcome sample space e.g. Throw a dice at trial i and record the outcome i. i {,,3,4,5,6} e.g. Measure the noise level o a phone line at time t and record the outcome (t. voltage (t (-0v, 0v Note that in e.g.,, the outcome (t is continuous, so there e are ininite number o possible outcome. What will be the probability o each o this outcome? The probability o a possible outcome is 0. ( 3 may not be 0.
4 5. Continuous Random Variables (4 De: Let represent a continuous random variable in a sample space S. or each we have a probability density unction (p.d.. ( which h is a unction that t satisies the ollowing: ( 0 S p( ( d s 3 in S, the probability o [, ] ( ( d I is continuous at a, then (=a, i.e. the probability o being a a a The 3 aioms o robability. 0 (A ( d. (S = 3. AB = Ø (AB= (A+ (B 0
5 Why do we need probability density unction? (5 Quantizer discrete output t continuous input ( Gaussian distribution σ ep σ Laplacian distribution ep
6 How to determine the p.d.. o a random variable by eperiments? (6 (t ( t t t3 t 4 t T ( choose a value or T (accuracy improves as T ( choose a value or (resolution improves as ( d ( or, ( will be approimately constant over that interval ( ( (, [, ] ( (, [, ] ( i j t T i
7 Cumulative Distribution unction (7 De: The cumulative distribution unction or distribution unction ( o a random variable is the probability that is less than or equal to, i.e. ( ( ( d. e.g. ( ½ ( ½
8 roperties o Distribution unction (8 0 ( ( ( distribution unction. 0. d ( ( ( ( d ( ( ( or ( ( ( 3. : These sets are mutually eclusively. i 4. ( e.g. i.e. ( is monotonically non-decreasing in. (
9 roperties o Distribution unction (9 d 5. i d is continuous. : d (prop.3 Consider the case when ( is continuous at = and = where is a small positive number. lim 0 d distribution unction ( ( + ( d lim 0 lim 0 d d 6. ( a b = ( a < b = ( a< b =( a< < b or any a and b where a b because ( =a = ( =b =0 so it does not matter whether either endpoint is included.
10 e.g. (chapter 5 eercises 7 & 8 The distribution unction o a random variable is given by 0 or. or distribution unction ( ( ( d (a indthe probabilities that this random variable will take on a value less than 3; and between 4 and 5; (b ind the p.d.. (. Are there any points at which ( is undeined? (c plot ( and (. sln. (a (<3 = (3 = 8 / 9 = (4 5 = (5 - (4 = / 6 - / 5 =
11 e.g. (chapter 5 eercises 7 & 8 I the distribution unction o a or 0 or random variable is given by. (a ind the probabilities that this random variable will take on a value less than 3; and between 4 and 5; (b ind the p.d.. (. Are there any points at which ( is undeined? (c plot ( and (. Sln. (continued (b ( d d (c ( is not dierentiable at = and so ( is undeined at =.
12 ( ( Mean o probability density ( ( d Mean o probability distribution ( ( all Variance o probability density ( ( ( d Variance o probability distribution ( all ( (
13 kthmoment about the origin ' k k d k,,...,... Mean o probability distribution ( all ( Mean o probability density ( ' st ( moment about d the origin k th moment about the mean Variance o probability bilit distribution ib ti ( k k ( d k,,... all ( ( Variance o probability density ( ( ( d Computing ormula or the variance ' nd moment about the mean
14 rove: sln ' ( ( d ( d ( d ( d k th moment about the origin ' k k ( d k th moment about the mean k k ( ( d ' ' average power o A.C. component average power o A.C.+ D.C. power o D.C. component - t t t
15 5. The Normal Distribution (5 Normal distribution ( ;, e σ ( d ( ( d Normal distribution, also known as Gaussian distribution, is the most commonly used pd p.d..
16 The normal distribution cannot be integrated in closed orm between every ypair o limits a and b. The probability relating to normal distributions are obtained rom special tables, such as standard normal distribution (=0, = table. Its entries are the values o the cumulative distribution unction z ( z e t dt ( Z z which is the area under standard normal ds distribution to the let o z. (t distribution unction z ( z (Z z ( d normal distribution ( ;, σ e standard normal (z ( z e t z distribution
17 To ind the probability that a random variable having the standard normal distribution (=0, = will take on a value between a and b, i.e. ( a< Z b, we use (b - (a. standard normal (z ( z e z distribution z (t (z t eg e.g. A random variable has the standard normal distribution. ind (a ( 0.87< Z.8 = (.8 - (0.87 = = (b ( 0.85< Z = - (0.85 = = standard normal distribution table
18 standard normal distribution table (=0, = (z = (Z Z z (0.85 (087 (0.87 (.8
19 The z notation or a standard normal distribution (z z is the value o z such that ( z < Z = (z ( z ( Z z e.g. ind (a z 0.0 ; (b z sln. (a (z 0.0 = = 0.99 rom the standard normal distribution table, z 00 =.33. z, 0.0 (b (z 0.05 = = 0.95 rom the standard normal distribution table, z 0.05 =.645. standard normal distribution table
20 standard normal Distribution table (=0, = (z = (Z Z z (z z (z 0.0
21 To use a standard normal distribution table in connection with a random variable which has a normal distribution with the mean and the variance, we reer to the ollowing standardized random variable, Z which can be shown to have the standard normal distribution. When has the normal distribution with normal distribution mean and standard variance, b a ( a b. ( ;, σ e standard dnormal distributi ib tion ( z e z
22 e.g. The actual amount o instant coee that a illing machine puts into 4-ounce jars may be looked upon as a random variable having a normal distribution with 0.04 ounce. I only % o the jars are to contain less than 4 ounces, what should be the mean ill o these jars? sln. Let Z where To ind such that ( z 0.0, 0.04 (z=0.0 z we look or the entry in a standard normal distribution table closest to 00and 0.0 get 0.008, corresponding to z = Hence, The mean = = 4.08 standard normal distribution table
23 z standard normal Distribution table (=0, = (z = (Z Z z z (-.05
24 e.g. In a city, the number o power outages per month is a random variable, having a distribution with.6 and 3.3. I this distribution can be approimated closely l with a normal distribution, ib ti what is the probability bilit that there will be at least 8 outages in any one month? sln. The answer is given by the area o the shaded region the area to the right o 7.5 instead o 8. It is because the number o outages is a discrete random variable. I we want to approimate its distribution with a normal distribution, we must spread the discrete values over the continuous scale We represent each integer k by the interval k- / and k+ /. or eample, 8 is represented by the interval 7.5 to (.4 = = standard normal distribution table
25 z standard normal Distribution table (=0, = (z = (Z z Number outages
26 5.3 The Normal Approimation to the Binomial Distribution (6 Theorem : I is a random variable having the binomial distribution with the parameters n and p, the limiting orm o the distribution unction o the standardized random variable np Z np( p as n, is given by the standard normal distribution ( z z e t dt z Standardized random variable b Z binomial distribution n ( ; n, p Cn, p ( p binomial distribution = np = np (-p
27 e.g. I 0% o the memory chips made in a certain plant are deective, what are the probabilities that in a lot o 00 randomly chosen or inspection (a at most 5 will be deective; (b eactly 5 will be deective? sln. p = 0.; n = 00 so = 00 0.=0 and = = 6. binomial distribution = np (a at most 5 will be deective = np (-p 5.5 ( (b eactly 5 will be deective By binomial distribution 0.85 We represent discrete RV k by the interval o continuous RV k - / & k + / (.3 (.38 = = By binomial distribution A good rule o thumb or the normal approimation Use the normal approimation to the binomial distribution only when Use the normal approimation to the binomial distribution only when np and n(-p are both greater than 5.
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