Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Statistics in Geophysics: Probability Theory II Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Random variables In many experiments it is easier to deal with a summary variable than with the original probability structure Example: In an opinion poll, we might to decide to ask 50 people whether they agree or disagree with a certain issue The sample space for this experiment has 250 elements Define a variable X = number of 1s recorded out of 50 The sample space for X is the set of integers {0, 1, 2, , 50} Winter Term 2013/14 2/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Random variables Definition: A random variable is a function from a sample space Ω into the real numbers We have also defined a new sample space (the range of the random variable) Suppose we have a sample space Ω = {ω1 , , ωn } with a probability function P and we define a random variable X with range X = {x1 , , xm } Winter Term 2013/14 3/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Random variables We define an induced probability function PX on X as follows: PX (X = xi ) = P({ωj ∈ Ω : X (ωj ) = xi }) , for i = 1, , m and j = 1, , n We will simply write P(X = xi ) rather than PX (X = xi ) Winter Term 2013/14 4/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Example: Tossing a fair coin three times X : number of heads obtained in the three tosses ω X (ω) HHH HHT HTH THH TTH THT HTT TTT Table: Enumeration of the value of X for each point in the sample space x P(X = x) 1/8 3/8 3/8 1/8 Table: Induced probability function on X For example, P(X = 1) = P({HTT,THT,TTH}) = 38 Winter Term 2013/14 5/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Distribution function Definition: The cumulative distribution function or cdf of a random variable X , denoted by FX (x), is defined by FX (x) = P(X ≤ x), for all x Example (Tossing three coins): The cdf of X is if ∞ < x < if ≤ x < 1 FX (x) = if ≤ x < if ≤ x < if ≤ x < ∞ Winter Term 2013/14 6/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Properties of a cdf The function FX (x) is a cdf if and only if the following three conditions hold: lim x→−∞ FX (x) = and limx→∞ FX (x) = FX (x) is a monotone, non-decreasing function of x FX (x) is continuous from the right; that is, lim FX (x + h) = FX (x) 0 Winter Term 2013/14 40/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Independence Definition: Let (X , Y ) be a bivariate random vector with joint pdf or pmf fX ,Y (x, y ) and marginal pdfs or pmfs fX (x) and fY (y ) Then X and Y are called independent if, for every x, y ∈ R, fX ,Y = fX (x)fY (y ) If X and Y are independent, then fX |Y (x|y ) = fX (x) fY |X (y |x) = fY (y ) and Winter Term 2013/14 41/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Expectation Definition: The expected value of a function g (X , Y ) of the random vector (X , Y ), denoted by E(g (X , Y )), is E(g (X , Y )) = g (x, y )fX ,Y (x, y ) , if (X , Y ) is discrete, where the summation is over all possible values of (X , Y ), and ∞ ∞ −∞ −∞ E(g (X , Y )) = g (x, y )fX ,Y (x, y ) dx dy , if (X , Y ) is continuous Winter Term 2013/14 42/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Covariance and correlation The covariance of X and Y is the number defined by Cov(X , Y ) = E[(X − µX )(Y − µY )] = E(XY ) − µX µY The correlation of X and Y is the number defined by ρXY = Cov(X , Y ) , σX σY −1 ≤ ρXY ≤ The value ρXY is also called the correlation coefficient X and Y are called uncorrelated if ρXY = 0; they are positively (negatively) correlated if ρXY > (ρXY < 0) Winter Term 2013/14 43/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Properties of covariance and correlation The following statements hold: If X and Y are independent random variables, then Cov(X , Y ) = and ρXY = If X and Y are any two random variables and a and b are any two constants, then Var(aX + bY ) = a2 Var(X ) + b2 Var(Y ) + 2ab Cov(X , Y ) If X and Y are independent random variables, then Var(aX + bY ) = a2 Var(X ) + b2 Var(Y ) Winter Term 2013/14 44/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Example: The bivariate standard normal distribution The bivariate standard normal distribution with parameter ρ (|ρ| < 1) has the joint density f (x, y ) = 2π − ρ2 exp − (x − 2ρxy + y ) (1 − ρ2 ) The marginal distributions of X and Y are (for any ρ) standard normally distributed The correlation of X and Y is ρ In this case: Uncorrelatedness implies independence Winter Term 2013/14 45/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Example: The bivariate standard normal distribution ● ● ● 3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.18 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −3 −3 ● ● ● ● ● ● −1 ● −3 ● −2 ● ● ●● ● ● ● −3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.06 ● ● ● ● ● 0.16 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● 0.1 ● ● ● ● ● ● ●● ● ● ● 0.14 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.12 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.08 ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.22 0.2 ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● 0.04 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 0.08 ● ● ● ● ● ● 0.18 ●● ● ● ● ● ● ● ● 0.12 ●● ● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.02 ● ● ● ● ● ● ● ● ● ● 0.16 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ●● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.14 ●● ● ●● ●●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● 0.1 ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ●● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● 0.06 ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.14 ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.12 ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● ● ● ● ● ● ● ● ● ● ● y ● ● ● ● ● ● ● ● ● ● ● 0.04 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● −2 ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.06 0.08 ● ● ● ● ● ● ● y ● ● ● ● ● ● ● ● ● ● y 2 ● ● ● ● ● 0.04 ● ● ● ● ● ● ● ● ● ● ● ● 0.02 ● ● 0.02 ● ● ● ● ● ● ● ● ●● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● −3 −2 x −1 x ● −3 −2 −1 x Contour plots of the joint density of the bivariate standard normal distribution, obtained using 500 samples for ρ = (left), ρ = 0.7 (middle) und ρ = −0.5 (right) Winter Term 2013/14 46/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Sums of random variables If X and Y are independent random variables with pmfs or pdfs fX (x) and fY (y ), then the pmf or pdf of Z = X + Y is fX (z − y )fY (y ) , fZ (z) = P(X + Y = z) = y if X and Y are discrete and ∞ fX (z − y )fY (y ) dy , fZ (z) = −∞ if X and Y are continuous The function fZ (z) is called the convolution of fX (x) and fY (y ) Example: X ∼ N (µX , σX2 ), Y ∼ N (µY , σY2 ) and independent, then Z ∼ N (µX + µY , σX2 + σY2 ) Winter Term 2013/14 47/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Law of large numbers Consider independently and identically distributed (i.i.d) random variables X1 , X2 , , Xn with E(Xi ) = µ and Var(Xi ) = σ < ∞ (i = 1, , n) If we define n ¯ Xi , Xn = n i=1 ¯n ) = µ and Var(X ¯n ) = it can be shown that E(X The law of large numbers states that P for every ¯n − µ ≥ lim X n→∞ > Winter Term 2013/14 48/50 =0 , n σ2 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Central limit theorem A random variable X with mean µ = E(X ) and variance σ = Var(X ) can be linearly transformed, such that the ˜ has zero mean and unit variance: transformed variable X ˜ = X −µ X σ Then, ˜) = E(X ˜) = Var(X (E(X ) − µ) = , σ Var(X ) = σ2 Winter Term 2013/14 49/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Central limit theorem Consider i.i.d random variables X1 , X2 , , Xn with E(Xi ) = µ and Var(Xi ) = σ (i = 1, , n) For the sum Yn = X1 + X2 + + Xn it holds that E(Yn ) = n · µ and Var(Yn ) = n · σ For the standardised sum Yn − nµ Zn = √ =√ n·σ n n i=1 Xi − µ , σ it therefore holds that E(Zn ) = und Var(Zn ) = The central limit theorem states that a Zn ∼ N (0, 1) Yn ∼ N (n · µ, n · σ ) a Winter Term 2013/14 50/50 [...]... different choices of π Winter Term 2013/14 23/50 70 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Poisson distribution The Poisson distribution describes the number of events occurring in a certain time interval and so pertains to data on counts A random variable X is defined to have a Poisson distribution,... distributions! Winter Term 2013/14 19/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Binomial distribution The binomial distribution is based on the idea of a Bernoulli trial A random variable X is defined to have a Bernoulli distribution, denoted by X ∼ B(π), if X = 1 with probability π 0 with probability. .. Common Probability Distributions Multiple Random Variables Probability density function Definition: The probability density function (pdf) fX (x), of a continuous random variable X is the function that satisfies x FX (x) = fx (t) dt for all x −∞ Since P(X = x) = 0, P(a < X < b) = P(a < X ≤ b) = P(a ≤ X < b) = P(a ≤ X ≤ b) Winter Term 2013/14 11/50 Random Variables and Distribution Functions Common Probability. .. Consider a sequence of n identical, independent Bernoulli trials, X1 , X2 , , Xn , each with success probability π X1 , , Xn are independent if P(X1 = x1 , , Xn = xn ) = Winter Term 2013/14 n i=1 P(Xi 20/50 = xi ) Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Binomial distribution Let X count... Discrete Distributions Continuous Distributions Introduction Statistical distributions are used to model populations We usually deal with a family of distributions, which is indexed by one or more parameters Here, we catalog some of the frequently occurring probability laws and examine the assumed chance mechanisms that lead to their usage This presentation is by no means comprehensive in its coverage of statistical... cdf (right) for X ∼ E(λ) with different intensities λ Winter Term 2013/14 30/50 10 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Normal distribution The normal distribution plays a central role in statistics and has many applications A random variable X , is defined to be normally distributed, denoted... 1 and λ = 3 Winter Term 2013/14 25/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Example: Annual Hurricane Landfalls on the U.S coastline Figure: Histogram of annual numbers of U.S landfalling hurricanes for 1899-1998 (dashed), and fitted Poisson distribution with λ = 1.7 (solid) Winter Term 2013/14... Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Approximation of the binomial distribution 0.4 π = 0.5 0.4 π = 0.8 Poisson Binomial 0.3 0.3 Poisson Binomial ● ● ● 0.1 0.2 f(x) ● ● ● ● ● 0 ● ● ● 2 ● ● ● ● 4 ● ● ● 0.0 0.0 ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● 0.1 0.2 f(x) ● ● 6 8 10 ●● ● ● ● ● 0 2 4 6 x 8 ● ● 10 x 0.4 π = 0.1 0.4 π = 0.3 Poisson Binomial... Poisson Binomial 0.3 0.3 ● ● ● ● ● 0.2 f(x) 0.2 ● ● ● ● ● 0.1 ●● ● 0 ●● ● ● ● ● ● 2 4 6 ●● 8 ●● ●● 10 Winterx Term 2013/14 ●● 0.0 0.1 ● 0.0 f(x) ● ● ● 0 27/50 1 2 3 4 x ●● ●● ●● 5 6 7 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Discrete Distributions Continuous Distributions Approximation of the binomial distribution π = 0.8 π = 0.5 Poisson Binomial... such a statement by giving either the cdf or the pdf (or pmf) of the random variable of interest Winter Term 2013/14 13/50 Random Variables and Distribution Functions Common Probability Distributions Multiple Random Variables Mean of a random variable Definition: Let X be a random variable The expected value or mean of X , denoted by E(X ) (or µX ), is (provided that the sum or integral exists): (i) ... heads obtained in the three tosses ω X (ω) HHH HHT HTH THH TTH THT HTT TTT Table: Enumeration of the value of X for each point in the sample space x P(X = x) 1/8 3/8 3/8 1/8 Table: Induced probability. .. distribution describes the number of events occurring in a certain time interval and so pertains to data on counts A random variable X is defined to have a Poisson distribution, denoted by X... Common Probability Distributions Multiple Random Variables Random variables In many experiments it is easier to deal with a summary variable than with the original probability structure Example: In