Week 9, 10/15/12-10/19/12, Notes: Continuous Distributions in General and the Uniform Distribution

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1 Week 9, 10/15/12-10/19/12, Notes: Continuous Distributions in General and the Uniform Distribution 1 Monday s, 10/15/12, notes: Review Review days are generated by student questions. No material will be distributed for this day. See lecture notes, extra examples, and past tests for possible questions to ask. 2 Wednesday s, 10/17/12, notes: Continuous Distributions in General A continuous random variable typically involves measurement. One way to define a continuous random variable is that it has no point mass, or no point probabilities. This is in direct contrast to discrete random variables. Mathematically, a random variable X is called a continuous r.v. if P(X=x) = 0 for all x in R. Some useful set notation is that x (0,1) is {x: 0 < x < 1} while x [0,1] means {x: 0 x 1}. Cumulative Distribution Function, cdf, is a key topic for r.v.s (discrete and continuous alike). Let X be a r.v., the the cdf of X, denoted by F X (x) is the real-valued function defined on R by F X (x) = P (X x) such that x is in R. While a cdf applies to any type of r.v., we typically only use it with respect to continuous r.v.s. The reason for this is that most discrete random variables do not have a nice functional form for their cdf. Example 9.1a Let us find the cdf of a coin tossing example. number of heads in the sample. Find the cdf for X. Let n=4, p=.3, and X be the Example 9.1b Keep the above set-up, but use p=.5 instead. What is the cdf for this r.v.? Example 9.2a Let X denote a number selected at random from the interval (0,1), what is the cdf of X? Example 9.2b Let X denote a number selected at random from the interval (0,10), what is the cdf of X? Properties of a cdf 1. It is nondecreasing. 2. It is everywhere right-continuous.

2 3. It has a value of 0 for x = - 4. It has a value of 1 for x = Useful Identities 1. P(a < X b) = F X (b) F X (a) 2. P(a X b) = F X (b) F X (a ) 3. P(a < X < b) = F X (b ) F X (a) 4. P(a < X < b) = F X (b ) F X (a ) Most of the above are really important when we have a cdf that has a jump (whether it is a cdf for a discrete r.v. or a mixed r.v.). However, the idea of the probability of being in a region for a CONTINUOUS r.v. is the cdf at the higher x value minus the cdf at the lower x value. Putting this another way, F X (b ) = F X (b) and F X (a ) = F X (a) for all values of a and b if X is a continuous r.v. Probability Density Function, pdf is another key topic for continuous r.v.s. Let X be a continuous r.v. A nonnegative function f X is said to be a pdf for X if, for all real numbers a < b, P (a X b) = b a f X (x)dx The pdf is the derivative of the cdf (only where the cdf is nonzero. Anywhere the cdf is 0, the pdf is also 0.) Revisit examples 9.2a and 9.2b. What are the pdfs for these 2 problems? Properties of the pdf: 1. f X (x) 0 for all real numbers x. 2. f X(x)dx = P (a X b) = b a f X(x)dx for all real numbers a and b such that a b. Recall, item 3 above can also be written as F X (b) F X (a). This brings us back to the definition or formulation of the cdf. We can define the cdf in 2 ways. The first is more of the interpretation of the cdf and the second is how to calculate or find it, if it is not given in a problem. F X (x) = P (X x) F X (x) = x f X (u)du

3 Expected Value is still a big topic for continuous r.v.s. The formula is similar to that for discrete r.v.s. How do you think the sum would change for a continuous r.v.? How do you think p X (x) would change? E[X] =? Again, you can do general expectations for functions of a random variable. For any function of x, say g(x), you can find the expectation of g(x). E[g(x)] =? An intersting note is that not all continuous distributions have a finite expected value (sometimes they are infinite). If they do not have a finite expected value, we say they do not have an expected 1 value. A famous example is the Cauchy distribution, which has a pdf of which takes values π(1+x 2 ) anywhere in R. Linearity Property of Expected Value Let X and Y be continuous r.v.s with a joint pdf and finite expectations. Also, let a, b, and c be real numbers. Then the following hold: 1. The random variable X + Y has finite expectation and E[X + Y] = E[X] + E[Y]. 2. E[cX] = c*e[x] 3. E[aX + by] = a*e[x] + b*e[y] 4. E[a + bx] = a + b*e[x] 5. if X Y, then E[X] E[Y] The distribution of a continuous r.v. X is said to be symmetric about a number θ if f X (x θ) = f X (θ x) for all values of x. If X is a continuous random variable such that E[X] exists and X is symmetric about θ, then E[X] = θ. Recall there are 2 different definitions of variance. and V ar(x) = E[(X E[X]) 2 ] V ar(x) = E[X 2 ] (E[X]) 2 Remember, the first definition is more about the interpretation of variance, and the second definition is usually a bit easier computationally. Percentiles and Special Percentiles A quartile represents a quarter of a data set or a quarter of a distribution. There are 3 quartiles of importance to a statistician (1 st, 2 nd, and 3 rd ). Sometimes the first and third quartiles are referred to as the lower and upper quartiles respectively.

4 The first quartile, Q1, represents the bottom (lower) 25% of the data. The second quartile, Q2, aka the median, represents the bottom (lower) 50% of the data. The third quartile, Q3, represents the bottom (lower) 75% of the data. Q1 is the x value for which F X (x) =.25. You can define similarly Q2 and Q3. A percentile represents the lower such-and-such percent of the distribution. For example, the 10 th percentile means that 10% of the distribution is that value, or it is the x-value such that F X (x) =.10. You can similarly define any other percentiles. Note: The quartiles are really just special cases of percentiles, especially the median. Example 9.3 Let X represent the diameter in inches of a circular disk cut by a machine. Let f X (x) = c(4x x 2 ) for 1 x 4 and be 0 otherwise. Answer the following questions: (a) Find the value of c that makes this a valid pdf. (b) Find the expected value and variance of X. (c) What is the probability that X is within.5 inches of the expected diameter? (d) Find F X (x). (e) What is the 33 rd percentile of X? Example 9.4 Let f X (x) =.25x for 1 x 3 and 0 otherwise. (a) Is X more likely to be within [1,2] or within [2,3]? First answer this question using logic. Next, check your answer by calculating the probabilities. (b) What is the probability that X is more than 2.2? (c) Find the mean and standard deviation of X. (d) Find F X (x). (e) What value of X represents the top 15% of the distribution? 3 Friday s, 10/19/12, notes: Uniform Distribution As the name implies, it has a uniform characteristic, this applies to its pdf. It is sometimes said to be evenly or uniformly distributed over an interval. This is a good way to characterize the distribution. Characteristics of the Uniform Distribution: The definition of X. The support is: Its parameter(s) and definition(s):

5 The pdf is: The cdf is: The expected value is: The variance is: Example 9.5 Revisit examples 9.2a and 9.2b. These examples are actually uniform distributions. Calculate the expected values and variances for these 2 distributions. Also, calculate the 41 st percentiles. Example 9.6 Shaggy feeds Scooby a Scooby-snack after every hi-jinks that Scooby foils. Suppose Scooby foils a hi-jinks anywhere from 0 minutes into the show up until 15 minutes into the show. Find the pdf, cdf, expected value, and variance for the amount of time until Scooby receives a Scooby-snack (denoted by X). Additionally, calculate the following probabilities: P(X < 5), P(X > 10), P(3 < X < 11), and P(X < 12 X > 4). Example 9.7 A very famous, always crowded restaurant names Shenanigans has porterhouse meal as its advertised special on Sweetest Day. It takes between 7 and 16 minutes to cook the porterhouse. Find the pdf, cdf, expected value, and variance for the amount of time until your porterhouse is cooked (denoted by X). Additionally, calculate the following probabilities: P(X < 10), P(X > 12), P(9 < X < 11), and P(X < 14 X > 11). Example 9.8 Suppose it takes Landfill between 4 seconds and 15 seconds to finish any given drink. Keep in mind that he has to deal with the noise coming from the glockenspiel. Let X be the amount of time it takes Landfill to finish his next drink. Name the distribution and parameter(s) of X. Find the probabilities that X is more than 8, less than 12, and between 8 and 12. Example 9.9 Anywhere from 0 to 20 years a really ridiculous political term gets added to the English dictionary. Examples include antidisestablishmentarianism, gerrymandering, and filibuster. What is the probability that the next quirky political term gets added to the dictionary sometime in the next 8 years? What about at least 13 years from now?