Class Notes (808,703)
Canada (493,376)
Statistics (237)
STAT151 (146)


4 Pages
Unlock Document

University of Alberta
Paul Cartledge

6.1 Random Variables Def’n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variableis a random variable that assumes separate values. e.g. # of people who think stats is dry The probability distribution of a discrete random variable lists all possible values that the random variable can assume and their corresponding probabilities. Notation: X = random variable; x = particular value; P(X = x) denotes probability that X equals the value x. Ex6.1) Toss a coin 3 times. Let X be the number of heads. 8 possible values: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT 6Xa0le x P(X = x) 0 0.125 1 0.375 2 0.375 3 0.125 Two noticeable characteristics for discrete probability distribution: 1. 0 ≤ P(X = x) ≤ 1 for each value of x 2. ∑ P(X = x) = 1 Ex6.2) “no heads”: P(X = 0) = 0.125 “at least one head”: P(X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3) = 0.375 + 0.375 + 0.125 = 0.875 “less than 2 heads”: P(X < 2) = P(X = 0) + P(X = 1) = 0.125 + 0.375 = 0.500 The population mean µ of a discrete random variable is a measure of the center of its distribution. It can be seen as a long-run average under replication. More precisely, µ = x P(X = x ) ∑ i i Sometimes referred to as µ = E(X) = the expected value of X (see STAT 265). Keep in mind that µ is not necessarily a “typical” value of X (it’s not the mode). Ex6.3) Using Table 6X0, µ = ∑ x i(X = x i = (0)(0.125) + (1)(0.375) + (2)(0.375) + (3)(0.125) = 1.5 Æ On average, the number of heads from 3 coin tosses is 1.5. Ex6.4) Toss an unfair coin 3 times (hypothetical). Let X be as in previous example. x P(X = x) µ = ∑ xiP(X = x i 0 0.10 1 0.05 = (0)(0.10) + (1)(0.05) + (2)(0.20) + (3)(0.65) 2 0.20 = 2.4 3 0.65 As 2 example shows, interpretation of µ as a measure of center of a distribution is more useful when the distribution is roughly symmetric, less useful when the distribution is highly skewed. The population standard deviation σ of a discrete random variable is a measure of variability of its distribution. As before, the standard deviation is defined as the square root of the population variance σ , given by 2 σ = ∑ (xi− µ) P(X = x ) i ∑ xiP(X = x ) i µ 2 Continuous Distribution: Def’n: A continuous random variable assumes any value contained in one or more intervals. e.g. average alcohol intake by a student, average alcohol outtake by a student The probability distributionof a continuous r.v. is specified by a curve. Two noticeable characteristics for continuous probability distribution (sans calculus): 1. The probability that X assumes a value in any interval lies in the range 0 to 1. 2. The interval containing all possible values has probability equal to 1, so the total area under the curve equals 1. (corresponding diagrams drawn in class) Using probability symbols, point 1 is denoted by P(a ≤ X ≤ b) = Area under the curve from a to b (diagram drawn in class) The probability that a continuous random variable X assumes a single value is always zero. This is because the area of a line, which represents a single point, is zero. (diagram drawn in class) In general, if a and b are two of the values that X can assume, then P(a) = 0 and P(b) = 0 Hence, P(a ≤ X ≤ b) = P(a < X < b). For a continuous probability distribution, the probability is always calculated for an interval. Either like abov
More Less

Related notes for STAT151

Log In


Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.