Random Variables and Probability Distributions PDF

Random Variables and Probability Distributions Stat 115 Forman Christian College (A Chartered University) Lahore Random Variables :A random variable is a variable whose values are determined by chance....

Random Variables and Probability Distributions Stat 115 Forman Christian College (A Chartered University) Lahore Random Variables :A random variable is a variable whose values are determined by chance. A random variable that assumes countable values is called a discrete random variable. For example: Discrete random variable 1. The number of cars sold at a dealership during a given month 2. The number of houses in a certain block 3. The number of fish caught on a fishing trip Random variable A random variable that can assume any value contained in one or more intervals is called a continuous random Continuous random variable variable. For example: 1. The length of a room 2. The time taken to commute from home to work 3. The weight of a fish 4. The price of a house 2  PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome. CHARACTERISTICS OF A PROBABILITY DISTRIBUTION 1. The probability of a particular outcome is between 0 and 1 inclusive. 2. The outcomes are mutually exclusive events. 3. The list is exhaustive. So the sum of the probabilities of the various events is equal to 1. Example Suppose we are interested in the number of heads showing face up on three tosses of a coin. This is the experiment. The possible results are: zero heads, one head, two heads, and three heads. What is the probability distribution for the number of heads? There are eight possible outcomes. A tail might appear face up on the first toss, another tail on the second toss, and another tail on the third toss of the coin. Or we might get a tail, tail, and head, in that order. We use the multiplication formula for counting outcomes (5–8). There are (2)(2)(2) or 8 possible results. Number of Probability of Outcomes Heads P(X) 0 1 0.125 8 1 3 0.375 8 2 3 0.375 8 3 1 0.125 8 Total 1 4  Practice Question The possible outcomes of an experiment involving the roll of a six‐sided die are a one‐ spot, a two‐ spot, a three‐spot, a four‐spot, a five‐spot, and a six‐spot. (a) Develop a probability distribution for the number of possible spots. (b) Portray the probability distribution graphically. (c) What is the sum of the probabilities? 5  Example 1: The following table presents the frequency Discrete Probabaility and relative frequency distributions of the number of Distribution vehicles owned by all 2000 families living in a small town. No. of vehicles Owned Frequency Relative Frequency X P(X) The probability distribution of a discrete 0 30 30/2000 =0.015 random variable lists all the possible 1 470 0.235 values that the random variable can assume and their corresponding 2 850 0.425 probabilities. 3 490 0.245 4 160 0.080 Total 2000 1 𝑃 𝑋 2 0.425 𝑃 𝑋 2 𝑃 𝑋 3 𝑃 𝑋 4 0.245 0.080 0.325 6 Example 2: The following table lists the probability distribution of the number of breakdowns per week for a machine based on past data. Breakdowns per week 0 1 2 3 Probability 0.15 0.20 0.35 0.30 (a) Find the probability that the number of breakdowns for this machine during a given week is i. exactly 2 ii. 0 to 2 iii. more than 1 iv. at most 1 (b) Find mean and Standard deviation. Solution: (a) i. 𝑃 𝑒𝑥𝑎𝑐𝑡𝑙𝑦 2 𝑏𝑟𝑒𝑎𝑘𝑑𝑜𝑤𝑛𝑠 𝑃 𝑋 2 0.35 ii. 𝑃 0 𝑡𝑜 2 𝑏𝑟𝑒𝑎𝑘𝑑𝑜𝑤𝑛𝑠 𝑃 0 𝑋 2 𝑃 𝑋 0 𝑃 𝑋 1 𝑃 𝑋 2 0.15 0.20 0.35 0.70 iii. 𝑃 𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 1 𝑏𝑟𝑒𝑎𝑘𝑑𝑜𝑤𝑛 𝑃 𝑋 1 𝑃 𝑋 2 𝑃 𝑋 3 0.35 0.30 0.65 iv. 𝑃 𝑎𝑡 𝑚𝑜𝑠𝑡 1 𝑏𝑟𝑒𝑎𝑘𝑑𝑜𝑤𝑛 𝑃 𝑋 1 𝑃 𝑋 0 𝑃 𝑋 1 0.15 0.20 0.35 7 Mean and Standardd eviation fo Disrecte random variables The mean of a discrete random variable x is the value that is expected to occur per repetition, on average, if an experiment is repeated a large number of times. It is denoted by 𝜇 and calculated as 𝜇 Σ𝑋𝑃 𝑋 The mean of a discrete random variable X is also called its expected value and is denoted by 𝐸 𝑋 ; that is 𝐸 𝑋 Σ𝑋𝑃 𝑋 The standard deviation of a discrete random variable x measures the spread of its probability distribution and is computed as σ Σ𝑋 𝑃 𝑋 𝜇 8 (b) Find mean and Standard deviation X 𝑷 𝑿 𝑿𝑷 𝑿 𝑿𝟐 𝑷 𝑿 0 0.15 0 0.15 0 0 0.15 0 1 0.20 1 0.20 0.20 1 0.20 0.20 2 0.35 2 0.35 0.70 2 0.35 1.4 3 0.30 3 0.30 0.90 3 0.30 8.1 Total Σ𝑋𝑃 𝑋 1.80 Σ𝑋 𝑃 𝑋 9.7 𝜇 Σ𝑋𝑃 𝑋 1.80 Thus, on average, this machine is expected to break down 1.80 times per week over a period of time. σ Σ𝑋 𝑃 𝑋 𝜇 9.7 1.8 9.7 3.24 6.46 2.54 9  Practcie Questions 1. A review of emergency room records at rural Millard Fellmore Memorial Hospital was performed to determine the probability distribution of the number of patients entering the emergency room during a 1‐hour period. The following table lists the distribution. Patients per hour 0 1 2 3 4 5 6 Probability 0.2725 0.3543 0.2303 0.0998 0.0324 0.0084 0.0023 Determine the probability that the number of patients entering the emergency room during a randomly selected 1‐hour period is i. 2 or more ii. exactly 5 iii. fewer than 3 iv. at most 1 2. One of the most profitable items at A1’s Auto Security Shop is the remote starting system. Let x be the number of such systems installed on a given day at this shop. The following table lists the frequency distribution of x for the past 80 days. X 1 2 3 4 5 f 8 20 24 16 12 a. Construct a probability distribution table for the number of remote starting systems installed on a given day. b. Are the probabilities listed in the table of part a exact or approximate probabilities of various out‐ comes? Explain. c. Find the following probabilities. i. P(x = 3) ii. P(x > 3) iii. P(2 < x < 4) iv. P(x