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Random Variables in Statistics

Learn about random variables, including discrete and continuous types, probability distributions, means, variances, and examples in various scenarios. Explore concepts with practical examples and diagrams.

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Random Variables in Statistics

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  1. Random Variable – A random variable is a variable whose value is a numerical outcome of a random phenomenon. – A random variable is a function or a rule that assigns a numerical value to each possible outcome of a statistical experiment. Two Types: 1. Discrete Random Variable – A discrete random variable has a countable number of possible values (There is a gap between possible values). 2. Continuous Random Variable – A continuous random variable takes all values in an interval of numbers.

  2. Examples Tossing a coin 3 times: Sample Space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Random variables : X1 = The number of heads. = {3, 2, 2, 2, 1, 1, 1, 0} X2 = The number of tails. = {0, 1, 1, 1, 2, 2, 2, 3}

  3. Rolling a Pair of Dice Sample Space:

  4. Rolling a Pair of Dice Random variable: X3 = Total no. of dots

  5. Rolling a Pair of Dice X4 = (positive) difference in the no. of dots

  6. Rolling a Pair of Dice X5 = Higher of the two.

  7. More Examples Survey: Random variables : X6 = Age in years. X7 = Gender {1=male, 0=female}. X8 = Height. Medical Studies: Random variables : X9 = Blood Pressure. X10 = {1=smoker, 0=non-smoker}.

  8. Probability Distribution Tossing a coin 3 times: Sample Space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Random variable : X1 = The number of heads. = {3, 2, 2, 2, 1, 1, 1, 0}

  9. Probability Histogram Tossing a coin 3 times: Random variable : X1 = The number of heads.

  10. Rolling a Pair of Dice Sample Space:

  11. Rolling a Pair of Dice Random variable: X3 = Total no. of dots

  12. Rolling a Pair of Dice Random variable: X3 = Total no. of dots 1. Pr(X3<5)= 2. Pr(3<X3<12)=

  13. Discrete Random Variable A discrete random variable X has a countable number of possible values. The probability distribution of X where, Every piis a between 0 and 1. p1 + p2 +…+ pk = 1.

  14. Mean of a Discrete R.V. The probability distribution of X Mean () = E(X) = x1p1+x2p2+…+ xkpk Variance (2) = V(X) = (x1-)2p1 + (x2-)2p2 + …+ (xk-) 2pk .

  15. Continuous Random Variable A continuous random variable X takes all values in an interval of numbers. Examples: X11 = Amount of rain in October. X12 = Amount of milk produced by a cow. X13 = Useful life of a bulb. X14 = Height of college students. X15 = Average salary of UWL faculty. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.

  16. Continuous Distributions • Normal Distribution • Uniform Distribution • Chi-squared Distribution • T-Distribution • F-Distribution • Gamma Distribution

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