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Joint Probability distribution. Joint Probability Distributions. Given two random variables X and Y that are defined on the same probability space, the Joint Distribution for X and Y defines the probability of events defined in terms of both X and Y .
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Joint Probability Distributions • Given two random variables X and Y that are defined on the same probability space, the Joint Distribution for X and Y defines the probability of events defined in terms of both X and Y. • In the case of only two random variables, this is called a Bivariate Distribution. • The concept generalizes to any number of random variables, giving a Multivariate Distribution.
Example • Consider the roll of a die and let A = 1 if the number is even (2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (2, 3 or 5) and B = 0 otherwise. Find the Joint Distribution of A and B? • P(A = 0, B = 0) = {1} = 1/6 • P(A = 0, B = 1) = {3, 5} = 2/6 • P(A = 1, B = 0) = {4, 6} = 2/6 • P(A = 1, B = 1) = {2} = 1/6
Discrete Joint Probability Distribution • X and Y are Independent • f(x, y) = f(x) x f(y) • X and Y are Dependent
Discrete Joint Probability Distribution • A die is flipped and a coin is tossed
Marginal Probability Distributions • Marginal Probability Distribution: the individual probability distribution of a random variable.
Discrete Joint Probability Distribution • A die is flipped and a coin is tossed
Joint Probability Distributions • X: the time until a computer server connects to your machine , Y: the time until the server authorizes you as a valid user. Each of these random variables measures the wait from a common starting time and X <Y. Assume that the joint probability density function for X and Y is • Find the probability that X<1000 and Y<2000.
Functions of Random Variables • Z = g(X, Y) • If we roll two dice and X and Y are the number of dice turn up in a trial, then Z = X + Y is the sum of those two numbers. Discrete Distribution Function Continuous Distribution Function
Addition of Means • The Mean (Expectation) of a sum of random variables equals the sum of Means (Expectations). • E(X1 +X2 + … + Xn ) = E(X1) + E(X2) + … + E(Xn)
Multiplication of Means • The Mean (Expectation) of the product of Independent random variables equals the product of Means (Expectations). • E(X1 X2 … Xn ) = E(X1) E(X2) … E(Xn)
Independence Two continuous random variables X and Y are said to be Independent, if fXY(x, y) = fx(X) fy(Y) for all x and y
Addition of Variances • The Variance of the sum of Independent random variables equals the sum of Variances of these variables. • Б2 = Б12 + Б22 -2 БXY • БXY = Covariance of X and Y = E(XY) - E(X) E(Y) • If X and Y are independent, then E(XY) = E(X) E(Y) • Б2 = Б12 + Б22
Problem 1 Let f(x, y) = k when 8 ≤ x ≤ 12 and 0 ≤ y ≤ 2 and zero elsewhere. Find k. Find P(X ≤ 11 and 1 ≤ Y ≤ 1.5) and P(9 ≤ X ≤ 13, and Y ≤ 1).
Problem 3 Let f(x, y) = k. If x > 0, y > 0, x +y < 3 and zero otherwise. Find k. Find P(X + Y ≤ 1) and P(Y > X).
Problem 7 What are the mean thickness and standard deviation of transfer cores each consisting of 50 layers of sheet metal and 49 insulating paper layers, if the metal sheets have mean thickness 0.5 mm each with a standard deviation of 0.05 mm and paper layers have mean thickness 0.05mm each with a standard deviation of 0.02 mm?
Problem 9 A 5-gear Assembly is put together with spacers between the gears. The mean thickness of the gears is 5.020 cm with a standard deviation of 0.003 cm. The mean thickness of spacers is 0.040 cm with a standard deviation of 0.002 cm. Find the mean and standard deviation of the assembled units consisting of 5 randomly selected gears and 4 randomly selected spacers.
Problem 11 Show that the random variables with the density f(x, y) = x + y and g(x, y) = (x+1/2)(y+1/2) If 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 and f(x, y) = 0 and g(x, y) = 0 and zero otherwise, have the same Marginal Distribution.
Problem 13 An electronic device consists of two components. Let X and Y [months] be the length of time until failure of the first and second components, respectively. Assume that X and Y have the Probability Density f(x, y) = 0.01e-0.1(x + y) If x > 0 and y > 0 and zero otherwise. a. Are X and Y dependent or independent? b. Find densities of Marginal Distribution. c. What is the probability that the first component has a lifetime of 10 months or longer?
Problem 15 Find P(X > Y) when (X, Y) has the Probability Density f(x, y) = 0.25e-0.5(x + y) If x ≥ 0, y ≥ 0 and zero otherwise.
Problem 17 Let (X, Y) have the Probability Function f(0, 0) = f(1, 1) = 1/8 f(0, 1) = f(1, 0) = 3/8 Are X and Y independent?
Marginal Probability Distributions • Example: For the random variables in the previous example, calculate the probability that Y exceeds 2000 milliseconds.
Conditional Probability Distributions • When two random variables are defined in a random experiment, knowledge of one can change the probabilities of the other.
Conditional Mean and Variance • Example: From the previous example, calculate P(Y=1|X=3), E(Y|1), and V(Y|1).
Independence • In some random experiments, knowledge of the values of X does not change any of the probabilities associated with the values for Y. • If two random variables are independent, then
Multiple Discrete Random Variables • Joint Probability Distributions • Multinomial Probability Distribution
Joint Probability Distributions • In some cases, more than two random variables are defined in a random experiment. • Marginal probability mass function
Joint Probability Distributions • Mean and Variance
Joint Probability Distributions • Conditional Probability Distributions • Independence
Multinomial Probability Distribution • A joint probability distribution for multiple discrete random variables that is quite useful in an extension of the binomial.
Multinomial Probability Distribution • Example: Of the 20 bits received, what is the probability that 14 are Excellent, 3 are Good, 2 are Fair, and 1 is Poor? Assume that the classifications of individual bits are independent events and that the probabilities of E, G, F, and P are 0.6, 0.3, 0.08, and 0.02, respectively. • One sequence of 20 bits that produces the specified numbers of bits in each class can be represented as: EEEEEEEEEEEEEEGGGFFP • P(EEEEEEEEEEEEEEGGGFFP)= • The number of sequences (Permutation of similar objects)=
Two Continuous Random Variables • Joint Probability Distributions • Marginal Probability Distributions • Conditional Probability Distributions • Independence
Conditional Probability Distributions • Example: For the random variables in the previous example, determine the conditional probability density function for Y given that X=x • Determine P(Y>2000|x=1500)
Conditional Probability Distributions • Mean and Variance
Conditional Probability Distributions • Example: For the random variables in the previous example, determine the conditional mean for Y given that x=1500
Independence • Example: Let the random variables X and Y denote the lengths of two dimensions of a machined part, respectively. • Assume that X and Y are independent random variables, and the distribution of X is normal with mean 10.5 mm and variance 0.0025 (mm)2 and that the distribution of Y is normal with mean 3.2 mm and variance 0.0036 (mm)2. • Determine the probability that 10.4 < X < 10.6 and 3.15 < Y < 3.25. • Because X,Y are independent
Multiple Continuous Random Variables • Marginal Probability
Multiple Continuous Random Variables • Mean and Variance • Independence
Covariance and Correlation • When two or more random variables are defined on a probability space, it is useful to describe how they vary together. • It is useful to measure the relationship between the variables.
Covariance • Covariance is a measure of linear relationship between the random variables. \ • The expected value of a function of two random variables h(X, Y ).
