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Chapter 7 Random Variables and Probability Distributions 1. Definition of random variable A random variable, X, is a numerical variable whose value depends on the outcome of a chance experiment. In more advanced mathematical treatments of probability, a random variable is defined as a function on a sample space, as follows: A random variable is a function that assigns a real number to each point in a sample space. Note: we generally denote random variables by capital letters (X, Y, etc) and denote the actual numbers taken by random variable by small letters (x, y, etc). 2. Two types of random variable According to their ranges, random variables are classified into two types: discrete r.v. and continuous r.v. Examples:

• The amount of rainfall at a particular location during the next year • The distance that a person throws baseball • The number of questions asked during a 1-hr lecture

3. Probability distributions for discrete r.v. The probability distribution for a r.v. is a model that describes the long-run behavior of the a random variable. For any discrete r.v. X, the function p(x)=P(X=x) for each x within the range of X is called the probability function of X. The set of values {x, p(x): x within the range of X} is called the probability distribution of X.

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Example: Let X = the number of courses for which a randomly selected students at York is registered. The probability distribution of X appears in the following table: What is the P(X=4)? What is the P(at least 5 courses)? Properties of discrete probability distributions: 0 ≤ p(x) ≤ 1 for every possible x value; Σ p(x) = 1 (the sum is over all values of x). Example: Check whether the function given by p(x)=(x+2)/25 for x=1,2,3,4,5 can serve as the probability distribution of a discrete random variable. 4. Probability distributions for continuous r.v. Example: If one looks at the distribution of the actual amount of water (in ounces) in “one gallon” bottles of spring water they might see something such as (128 ounces = 1 gallon = 3.784 liters)

x 1 2 3 4 5 6 7

p(x) 0.02 0.03 0.09 0.25 0.4 0.16 0.05

Probability histogram

Amounts rounded to nearest hundredth of an

ounce. E.g.128.00 ,127.49

Probability histogram

Amounts rounded to nearest ten thousands of

an ounce.

Limiting curve as the accuracy increases

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Definition: A probability distribution for a continuous random variable X is specified by a mathematical function denoted by f(x) which is called the density function. The graph of a density function is a smooth curve. The following requirements must be met:

1. f(x) ≥ 0 2. The total area under the density curve is equal to 1.

The probability that X falls in any particular interval is the area under the density curve that lies above the interval. Example: P(a<X<b) P(X<a) P(X>b) Note:

1. For a continuous r.v. the probability at each point is 0! Thus P(a < x < b) = P(a ≤ x < b) = P(a < x ≤ b) = P(a ≤ x ≤ b).

2. The probability that a continuous random variable x lies between a lower limit a and an upper limit b is

P(a < x < b) = (cumulative area to the left of b) – (cumulative area to the left of a) = P(x < b) – P(x < a) Example: The density function of a continuous r.v. Y is given by f(x) = 0.2 for 2<y<7 0 otherwise (a) Draw its graph and verify that the total area under the curve (above the x-axis) is equal to 1. (b) Find P(3<Y<5). (c) Find P(Y>5).