RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Transcription

1 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS A random variable(rv) is a variable (typically represented by x) that has a single numerical value for each outcome of an experiment. A discrete random variable has either a finite number of values or a number of values. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are, such as a number line. We will start with discrete random variables in the generic discrete and binomial distributions, but the rest of the semester will deal with continuous random variables in the normal distribution. A probability distribution is actually a model of a theoretically perfect population frequency distribution. A probability distribution is like a distribution based on data that behave perfectly, without the usual imperfections of samples. A probability distribution is a collection of values of a RV along with their corresponding probabilities. Probability distributions can be given in a table format, similar to a relative frequency table, or can be represented by a probability histogram (discrete) or a probability density curve (continuous). Vertical axis - probability Horizontal axis - random variable x Σ P(x) = 1 where the sum is done over all possible x's (RV's). 0 P(x) 1 for every value of x. Note that the probabilities are equal to their corresponding rectangular areas in the discrete probability histogram. This is because each rectangle has a width of one and a height equal to its corresponding probability. SO, THE TOTAL AREA UNDER ANY PROBABILITY HISTOGRAM OR DENSITY CURVE MUST BE 1. THIS CORRESPONDENCE BETWEEN AREA AND PROBABILITY IS EXTREMELY IMPORTANT IN STATISTICS. Binomial formula: 10% of people are left handed(lh). Let's create the probability distribution for x, where x represents the number of LH out of 2 RS people. What are the possible x's? x = 0, 1, or 2. P(x = 0) = P(R 1 and R 2 ) =.9x.9 =.9 2 =.81

2 P(x = 1) = P(L 1 and R 2 ) =.1x.9 =.09 P(x = 2) = P(L 1 and L 2 ) = P(L 1 L 2 ) =.1x.1 =.1 2 =.01 So, our probability distribution looks like: x P(x) But Σ P(x) 1. P(x = 1) = P(L 1 R 2 or R 1 L 2 ) = = 2 x.09 =.18 So, the distribution looks like: x P(x) So, if 5 people are RS, what is the probability that exactly 2 are LH? P(x = 2) P(L 1 L 2 R 3 R 4 R 5 ) b/c there are = 10 ways to arrange 2 LH and 3 RH. P(x = 2) = x P(L 1 L 2 R 3 R 4 R 5 ) = 10 x.1 2 x.9 3 =.0729 It turns out that = 5 C 2 b/c n C r is the same as DP when there are only 2 distin. items. P(x) = n C x p x q n-x where n = fixed number of trials x = the number of successes in n trials, so 0 x n. p = probability of success in any one of the n trials.

3 q = probability of failure in any one of the n trials. p = P(Success) = P(S) q = P(Failure) = P(F) = 1- p = the number of outcomes with exactly x successes and n-x failures among n trials. n C x p x q n-x = the probability of x successes among n trials for any one arrangement of x successes and n-x failures. THE STANDARD NORMAL DISTRIBUTION Discrete RV's - probability histogram Continuous RV's - probability density curve FOR A DENSITY CURVE DEPICTING THE DISTRIBUTION OF A CONTINUOUS RV, THE AREA UNDER THE CURVE IS 1, AND THERE IS A CORRESPONDENCE BETWEEN AREA AND PROBABILITY. µ = population mean σ = population standard deviation e π A normal distribution is a continuous probability distribution. Examples of continuous rv's are time, speed, height, length, weight. This is data rather than countable data. A continuous rv has an infinite number of values that can be represented by an interval on a number line. The graph of a normal distribution is called the normal curve. The normal distribution has the following properties: 1. mean, median, mode are all the same 2. normal curve is bell shaped and symmetric about the mean and actually fits an equation. Because e and π are constants, the normal curve's shape is determined completely by mean and standard deviation. µ determines the line of symmetry and σ determines the spread of the curve. A larger σ will result in a fatter and shorter curve. 3. Total area under the curve is equal to 1. Area under curve is intimately associated with probability. 4. Normal curve approaches, but never touches, the x-axis as it extends farther and farther away from mean. 5. Inflection points at µ - σ and µ + σ. Inflection point is where the concavity of the curve changes.

4 Between µ - σ and µ + σ, the curve is concave down. Left of µ - σ and right of µ + σ, the curve is concave up. 6. Empirical rule for bell shaped data: 68% of data lies within 1 standard deviation of the mean of data lies within 2 standard deviations of the mean 99.7% of the data lies with 3 standard deviations of the mean Standard normal curve has a mean µ = 0 and a standard deviation σ = 1. z-score is a standardized score for the rv x that represents how many standard deviations a raw score x is from the mean µ. z = The rv x is sort of a raw score, whereas z is a standardized version of the raw score. So, the number line under a standard normal curve is just the z score number line. Under a nonstandard normal curve will be the x number line and the corresponding z number line. Areas under the standard normal curve are easy for a computer to calculate and have been tabulated in the standard normal table ***** Note that the standard normal table gives only the probability corresponding to the area under the standard normal curve that is to the left of the vertical line above any specific z-score. ****** The second decimal place of the z-score is found across top row. So, a z-score of z = 1.58 has a corresponding probability of This is the area under the standard normal curve to the left of z = We can use our knowledge of symmetry along with the fact that the total area under the curve is 1, to calculate many other probabilities. IT IS ESSENTIAL TO AVOID CONFUSING Z SCORES AND AREAS. REMEMBER, A Z SCORE MEASURES THE NUMBER OF STANDARD DEVIATIONS THAT A VALUE IS AWAY FROM THE MEAN. Z SCORE: DISTANCE ALONG HORIZONTAL SCALE ON GRAPH; refer to the leftmost column and top row in table. AREA(or PROBABILITY): AREA UNDER THE CURVE; refer to the numbers in the body of table. Although a z score can be negative, the AREA under the curve (or the corresponding PROBABILITY) can NEVER be negative.

5 NOTATION: P(a < z < b) denotes the probability that the z score is BETWEEN a and b. P(z > a) denotes the probability that the z score is GREATER THAN a. P(z < a) denotes the probability that the z score is LESS THAN a. P(z = a) = 0 P( a < z < b) = P(a z b) The probability of getting a z score of at most b is equal to P(z b). The probability of getting a z score of at least b is equal to P(z b). It is important to interpret correctly key phrases such as at most, at least, more than, no more than, and so on. NONSTANDARD NORMAL DISTRIBUTIONS: FINDING PROBABILITIES Most normally distributed populations have a nonzero mean, a standard deviation different from 1, or both. These are called normal distributions. We are able to standardize nonstandard cases by transforming the nonstandard x's into standard z's. Procedure for finding probabilities for values of a RV with a nonstandard normal probability distribution: 1. Write the probability question down in terms of x. 2. Transform that question into a probability question in terms of z. 3. Draw a normal curve along with an x number line and a corresponding z number line beneath that, label the mean and any relevant x and z scores, then shade the region representing your desired probability. 4. Refer to the normal table to find the areas corresponding to the relevant z scores. Include these numbers above your diagram. 5. Use your knowledge of symmetry and the fact that the total area under the curve is 1 to calculate the area of the desired shaded region. This area is the desired probability. NONSTANDARD NORMAL DISTRIBUTIONS: FINDING SCORES In the previous section we used a given score to find a probability. In this section we follow the reverse procedure we use a given probability to find a score. 1. Starting with a rough sketch that bears at least some resemblance to a bell, enter the given probability (or percentage) in the appropriate area of the graph and thus identify the location of the x value being sought. 2. Identify the area to the left of this x location. 3. Use the normal table to find the z score corresponding to this area to the left. You will likely not find the exact area in the table, so you will find the area in the body of the table that is closest to the area in your diagram.

6 4. Enter the values for µ, σ, and the z score found in step 3 into the following formula. Based on the format of the z score formula, we can solve for x as follows, x = µ + (z σ ). In considering problems of finding scores when given probabilities, there are 3 important cautions to keep in mind. 1. Don t confuse z scores and areas. Remember, z scores are distances along the horizontal scale, but areas represent probabilities under the normal curve. The normal table lists z scores in the left column and across the top row, but areas are found in the body of the table. 2. Choose the correct (left/right) side of the graph. A score separating the bottom 10% from the others will be located on the left side of the graph, but a score separating the top 10% will be located on the right side of the graph. P 10 represents the 10 th percentile and only 10% of the data is below this score. Thus, P 10 is the x score to the far left of the normal curve. P 90 represents the 90 th percentile and 90% of the data is below this score. Thus, P 90 is the x score to the far right of the normal curve. So, in general, P n represents the nth percentile and n% of the data is below this score. Q 1 = P 25, Q 2 = P 50, Q 3 = P 75, are quartiles and D 1 = P 10, D 2 = P 20, etc are deciles. 3. A z score must be negative whenever it is located to the left of the centerline.