Probability Distribution of Discrete Random Variables PDF

PROBABILITY DISTRIBUTION OF DISCRETE RANDOM VARIABLES A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities.  Example 1: If two coins are tossed, the possible outcomes are HH, HT, TH, TT. If X is the random variable...

PROBABILITY DISTRIBUTION OF DISCRETE RANDOM VARIABLES A discrete probability distribution consists of the values a random variable can assume and the corresponding probabilities.  Example 1: If two coins are tossed, the possible outcomes are HH, HT, TH, TT. If X is the random variable for the number of heads. NO HEADS ONE HEAD TWO HEAD POSSIBLE VALUES OF OUTCOMES THE RANDOM VARIABLE X HH 2 HT 1 TH 1 Number of heads, 0 1 2 X TT 0 Probability, P(X) 1/4 1/2 1/4  Example 2: Construct a probability distribution for rolling a single die. Sample spaces are {1,2,3,4,5,6} Outcome, 1 2 3 4 5 6 X P(X) Properties of Discrete Probability Distribution 1. The sum of all probabilities should be 1. 2. Probabilities should be confined between 0 and 1. Number of 0 1 2 heads, X Probability, 1/4 1/2 1/4 P(X) Example 3: Determine whether the distributions is a discrete probability distribution. X 3 6 8 X 1 2 3 4 5 P(X) -0.3 0.6 0.7 P(X) 3/10 1/10 1/10 2/10 3/10 Example 4: Supposed three coins are tossed. Let Y be the random variable representing the number of tails. Construct the probability distribution and draw the histogram. REGIONS UNDER NORMAL CURVE DISTRIBUTION  The Z-Score Table: Z-score less than '0' signifies an element less than the mean. A z-score more than '0' symbolizes an element bigger than the mean. Z-score equivalent to '0' denotes an element that is equal to the mean. Z-score which is equal to '1' represents an element which is one standard deviation superior to the mean. When a z-score is equal to -1' it represents an element which is one standard deviation less than the mean. If the quantity of elements in the set is huge, hence, about 68 percent of all the elements have a z-score between '-1' and '1'. Moreover, about 95 percent will be having a z-score between '-2' and '2', and lastly, about 99 percent of them will be having a z-score between '-3' and '3'.  hundredth place of a z-values z-values Example 1: Find the area that corresponds z = 2. The area that corresponds z = 2 is 0.4772 (47.72%) Example 2: Find the area that corresponds z = 2.47 The area that corresponds z = 2.47 is 0.4932 (49.32%) Example 3: Find the area that corresponds z = -2.47 The area that corresponds z = -2.47 is 0.4932 (49.32%) Example 4: Find the area that corresponds below 1.87 1.87 = 0.4693 0.5000 + 0.4693 = 0.9693 = 96.93% Example 5: Find the area that corresponds above 0.21 0.21 = 0.0832 0.5000 - 0.0832 = 0.4168 = 41.68% Example 6: Find the area that corresponds between -2.23 and -0.49 -2.23 = 0.4871 -0.49 = 0.1879 0.4871 - 0.1879 = 0.2992 = 29.92% Example 7: Find the area that corresponds outside -2.00 and 1.00  -2.00 = 0.4772 1.00 = 0.3413 (0.5000-0.4772) = 0.0228 (0.5000-0.3413) = 0.1587 0.0228 + 0.1587 = 0.1815 = 18.15%  NORMAL DISTRIBUTIONS (The Empirical Rule) Normal Probability Distribution is a probability distribution of continuous random variables. Many random variables are either normally distributed or, at least approximately normally distributed. Examples: Height and weight PROPERTIES OF NORMAL DISTRIBUTIONS  The distribution curve is bell-shaped The curve is symmetrical about its center. The mean, median, and mode coincide at the center The tails of the curve flatten out indefinitely along the horizontal axis but never touch it. (the curve is asymptotic to the base 50% 50% line) The area under the curve is 1. thus, it represents the probability or proportion or the percentage associated with specific sets of measurement values  Factors that affect the curve of the normal distributions: The change of value of the mean shifts the graph of the normal curve to the right or to the left. The standard deviation determines the shape of the graphs (particularly the height and width of the curve). When the standard deviation is large, the normal curve is short and wide, while a small value for the standard deviation yields skinnier and taller graph. THE EMPIRICAL RULE The Empirical Rule is also referred to as the 68-95-99.7% Rule. What it tells us is that for a normally distributed variable, the following are true: Approximately 68% of the data lie within 1 standard deviation of the mean. Pr(μ-σ