A random variable X is a function that assigns (real) numbers to the elements of the sample space S of a random experiment.

Transcription

1 RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS A random variable X is a function that assigns (real) numbers to the elements of the samle sace S of a random exeriment. The value sace V of a random variable is the set of all ossible values of the r. v. X. A discrete random variable is one whose value sace is finite or countably infinite. A continuous r.v. is one whose value sace is an interval of real numbers. The robability function of a discrete random variable is the function that secifies the robabilities of the r.v. assuming each of the various values in the value sace. That is, ( x) P[ X x]. The robability distribution of a discrete random variable consists of its value sace together with its robability function. That is, the "ossibilities" together with their "robabilities". It may be secified in function form or may be resented as a table of values or as a robability histogram. EXAMPLE: A "hand" of cards is dealt from a thoroughly shuffled deck of cards. There are,98,960 different ossible hands in the samle sace S. Let X denote the number of "Hearts" in the hand so dealt. Then the value sace is V {0, 1,, 3, 4, }. (V lists the ossibilities) The Probabilities are: ( 0) P[ X 0] 0. 1 () 1 P[ X 1] Page 1 of 1

2 3 3 ( ) P[ X ] ( 3) P[ X 3] ( 4) P[ X 4] ( ) P[ X ] The robability function for this r.v. X is ( x) P[ X x] x x The robability distribution of this r.v. X is ( ) [ ] x x x P X x for x V ; i.e. for x 0,1,,3,4,. Probability Distribution in table form Possible Value: x Probability: (x) Total or Basic Requirements of (x) a. 0 ( x) 1 for each x ε V b. ( x) 1 all xεv Page of

3 Histogram of Probability Distribution Probability Histogram for Num ber of Hearts in -Card Hand Probability Number of Hearts (Value of X) Question: What is the average number of Hearts in a -card hand selected this way? Definition: The mean or exected value of a discrete random variable X is defined as [ X ] x.( x) µ E. Examle: µ E [ X ] x ( x) x 0 x ( x) ( 0 0.1) + ( ) + ( 0.743) + ( ) + ( ) + ( ) but is really 1. and is off because of rounding robabilities to 4 decimal laces. Page 3 of 3

4 Calculating the Mean in Table Form x (x) x (x) Total µ Definition: The Variance of a discrete r.v. X is σ Var [ ] [ X ] E ( X µ ) ( x µ ).( x) all x in V Examle: σ Var x 0 [ ] [ X ] E ( X µ ) ( x 1.).( x) ( 0 1.) ( 0.1) + ( 1 1.) ( ) + ( 1.) ( 0.743) + ( 3 1.) ( 0.081) + ( 4 1.) ( ) + ( 1.) ( 0.000) [Based on other information, the correct value without rounding error is ] Comment: The variance is the exected value of the quantity (X-µ). It is a measure of the amount of variability or variation to be exected among the ossible values of a random variable. Page 4 of 4

5 Definition: The exected value of X, the Square of X E [ X ] x ( x) Examle: E [ X ] x ( x) x 0 x Calculating E[ X ] ( x) in Table Form x (x) x (x) Total [ X ] E.48 Theorem: The variance of any random variable X can be determined by σ Var E [ ] [ X ] E ( X µ ) (the definition) [ X ] µ (a useful calculation method) Examle: E [ X ] µ.48 ( 1.498) so that 68 Actually, E[ X ] σ 16 [ X ] E µ 68 4 Page of

6 which is what we had stated earlier. The rounding error is larger here because of the squaring taking lace. Definition: The Standard Deviation of a r.v. X is the square root of its variance. [ X ] σ σ Var. Examle: σ σ Comment: The standard deviation is the most commonly used measure of variation or variability of the values of a random variable. Emirical Rule If the shae of a robability distribution is mound-shaed and fairly symmetric, then the amount of robability between: a. µ σ and µ + σ is about 0.68 b. µ σ and µ + σ is about 0.9 c. µ 3 σ and µ + 3σ is almost 1.00 For a discrete random variable, look at its histogram. The histogram for the above examle of Hearts in a -card hand is not symmetric but it does have a mound or high region. How well does the Emirical Rule aly in this case? µ 1. and σ The interval between µ σ and µ + σ here is from µ σ to µ + σ Thus, P[ µ σ X µ + σ ] P[ 0.3 X.18] P[ X 1 or ] which is quite close to the redicted value of Because the histogram is not symmetric and mound-shaed we do not exect the Emirical Rule to work very well.. Similarly, the interval between µ σ and µ + σ is from µ σ to µ + σ and P µ σ X µ + σ P 0.61 X P X 0 or 1 or or 3 [ ] [ ] [ ] This value is considerably higher than the Emirical Rule value of 0.9. Page 6 of 6

7 Another examle with a erfectly symmetric and quite mound-shaed distribution follows. Examle: If one tosses a coin 10 times and counts the number of Heads observed in the 10 tosses, the robability distribution of the random variable Y number of Heads in 10 tosses has the robability histogram given below. For this random variable, µ.00 and σ 1.8. Number of Heads in 10 Tosses of a Coin Probability Number of Heads (Value of X) Reading robabilities from the histogram as accurately as you can, check to see how well the Emirical Rule works in this case. Examle: A random variable X is defined as the number of accidents a randomly chosen Saskatchewan driver has in a one-year eriod. Using accident records maintained by SGI over the ast ten years, the robability distribution for r.v. X was determined to be as follows. Number x Probability P[X x] How many accidents does one exect a tyical Saskatchewan driver to have in a 1-month eriod? How much variability does one exect to observe about this exected number? Page 7 of 7