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5. (12 pt, 3 each) The empirical cumulative distribution function (CDF) of a sample z=zi, ....
1. Consider the following Cumulative Distribution Function (CDF) of random variable 0.41 1 t <3 0.78...
1. Consider the following Cumulative Distribution Function (CDF) of random variable 0.41 1 t <3 0.78 3 < t < 5 0.94 5t<7 F(t) = a. 4 Find P(T> 3); P(1.5 < T b. [3] Find E(3T +5) and V (3T5) 6); P(T < 5IT2)
3. The cumulative distribution function (CDF), of a particular brand of flash drive is Fr(t) 0 for t s5.0, Fr(t) (t-5.0) for 5.0<ts6.0, and Fr(t) 1 for t> 6.0. Here, time t has the units of yea...
3. The cumulative distribution function (CDF), of a particular brand of flash drive is Fr(t) 0 for t s5.0, Fr(t) (t-5.0) for 5.0<ts6.0, and Fr(t) 1 for t> 6.0. Here, time t has the units of years. a. Sketch Fr(t) versus t. Show proper axis labels, scales, and units. b. Express mathematically reliability RT(t) versus t for t2 0, and sketch it below Fr(t) to the same horizontal scale. Show proper axis labels, scales, and units. Express mathematically probability density...
Let random variables X and Y have the bi-variate exponential CDF (cumulative distribution function) : F(x,y)...
Let random variables X and Y have the bi-variate exponential CDF (cumulative distribution function) : F(x,y) = 1 - exp(-x) - exp(-y) + exp(-x-y-xy) Given x > 0, y>0 a) Determine the probability that 4 < X given that Y = 2 b) Determine the probability that 4 < X given that Y is less than or equal to 2
Problem 6. Consider a random variable X whose cumulative distribution function (cdf) is given by 0...
Problem 6. Consider a random variable X whose cumulative distribution function (cdf) is given by 0 0.1 0.4 0.5 0.5 + q if -2 f 0 r< 2.2 if 2.2<a<3 If 3 < x < 4 We are also told that P(X > 3) = 0.1. (a) What is q? (b) Compute P(X2 -2> 2) (c) What is p(0)? What is p(1)? What is p(P(X S0)? (Here, p(.) denotes the probability mass function (pmf) for X) (d) Sketch a plot...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if -1 y <0.5, 0.7 Fr (y)- y < 2, if 0.5 1 if y 2 2. (a) Draw a sketch plot of Fy (y) d) Find the probability mass function, fz(2), of Z -Y2 (e Find El2] and Var(Z) (f) Find El2-321 and Var(2-32). 13 marks]
Previous Problem: Determine values of the cumulative distribution function for the random variable in the previous...
Previous Problem:
Determine values of the cumulative distribution function for the random variable in the previous problem. 3. 2. The probability mass function below is defined for x 0, 1,2,3,.. fr 5 5 -56 What is the probability for each of the following expressions? a) P(X 2) b) P(XE 2) c) P(X> 2) d) P(X2 1)
3. (5 pts) For each of the listed functions, determine a formula for the derivative function....
3. (5 pts) For each of the listed functions, determine a formula for the derivative function. For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. For the latter three, use the limit definition, Pay careful attention to the function names and independent variables. It is important to be comfortable with using letters other than f and r. For...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x;...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x; m, 7) = where m is the location parameter, is the scale parameter and they are both constants. We will create a function to simulate draws of a Cauchy(m, 7) distribution in this exercise using the inversion method. (a) (4 points) Derive the cdf, and the inverse function of cdf for Cauchy(m, n). Describe a procedure to generate independent observations from a Cauchy(m, 7)...
The graph of f is shown to the right. The function F(z) is defined by F(z) = f f(t) dt for 0 x 4. a) Find F(0) and F(3). 2 b) Find F (1). c) For what value of z does F(z) have its maximum value? What...
The graph of f is shown to the right. The function F(z) is defined by F(z) = f f(t) dt for 0 x 4. a) Find F(0) and F(3). 2 b) Find F (1). c) For what value of z does F(z) have its maximum value? What is this maximum value? d) Sketch a possible graph of F. Do not attempt to find a formula for F. (You could, but it is more work than neces- sary.) -1
The graph...
2.34. Probability integral transformation. Consider a random variable X with cumulative function Fx(x), 0-x-00, Now define a new random variable U to be a particular function of X, namely, U = Fx(X)...
2.34. Probability integral transformation. Consider a random variable X with cumulative function Fx(x), 0-x-00, Now define a new random variable U to be a particular function of X, namely, U = Fx(X) For example, if FX(x)-1-e-Ax, then U = 1-e-Ax = g(X). Show [at least for reasonably smooth Fx(x)] that the random variable U has a constant density function on the interval O to 1 and is zero elsewhere. Hint: Con vince yourself graphically thatgg (u)- u and assume that...
1. Consider the following Cumulative Distribution Function (CDF) of random variable 0.41 1 t <3 0.78 3 < t < 5 0.94 5t<7 F(t) = a. 4 Find P(T> 3); P(1.5 < T b. [3] Find E(3T +5) and V (3T5) 6); P(T < 5IT2)
3. The cumulative distribution function (CDF), of a particular brand of flash drive is Fr(t) 0 for t s5.0, Fr(t) (t-5.0) for 5.0<ts6.0, and Fr(t) 1 for t> 6.0. Here, time t has the units of years. a. Sketch Fr(t) versus t. Show proper axis labels, scales, and units. b. Express mathematically reliability RT(t) versus t for t2 0, and sketch it below Fr(t) to the same horizontal scale. Show proper axis labels, scales, and units. Express mathematically probability density...
Problem 6. Consider a random variable X whose cumulative distribution function (cdf) is given by 0 0.1 0.4 0.5 0.5 + q if -2 f 0 r< 2.2 if 2.2<a<3 If 3 < x < 4 We are also told that P(X > 3) = 0.1. (a) What is q? (b) Compute P(X2 -2> 2) (c) What is p(0)? What is p(1)? What is p(P(X S0)? (Here, p(.) denotes the probability mass function (pmf) for X) (d) Sketch a plot...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if -1 y <0.5, 0.7 Fr (y)- y < 2, if 0.5 1 if y 2 2. (a) Draw a sketch plot of Fy (y) d) Find the probability mass function, fz(2), of Z -Y2 (e Find El2] and Var(Z) (f) Find El2-321 and Var(2-32). 13 marks]
Previous Problem:
Determine values of the cumulative distribution function for the random variable in the previous problem. 3. 2. The probability mass function below is defined for x 0, 1,2,3,.. fr 5 5 -56 What is the probability for each of the following expressions? a) P(X 2) b) P(XE 2) c) P(X> 2) d) P(X2 1)
3. (5 pts) For each of the listed functions, determine a formula for the derivative function. For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. For the latter three, use the limit definition, Pay careful attention to the function names and independent variables. It is important to be comfortable with using letters other than f and r. For...
Question 3 A more general form of Cauchy distribution is defined by the density function f(x; m, 7) = where m is the location parameter, is the scale parameter and they are both constants. We will create a function to simulate draws of a Cauchy(m, 7) distribution in this exercise using the inversion method. (a) (4 points) Derive the cdf, and the inverse function of cdf for Cauchy(m, n). Describe a procedure to generate independent observations from a Cauchy(m, 7)...
The graph of f is shown to the right. The function F(z) is defined by F(z) = f f(t) dt for 0 x 4. a) Find F(0) and F(3). 2 b) Find F (1). c) For what value of z does F(z) have its maximum value? What is this maximum value? d) Sketch a possible graph of F. Do not attempt to find a formula for F. (You could, but it is more work than neces- sary.) -1
The graph...
2.34. Probability integral transformation. Consider a random variable X with cumulative function Fx(x), 0-x-00, Now define a new random variable U to be a particular function of X, namely, U = Fx(X) For example, if FX(x)-1-e-Ax, then U = 1-e-Ax = g(X). Show [at least for reasonably smooth Fx(x)] that the random variable U has a constant density function on the interval O to 1 and is zero elsewhere. Hint: Con vince yourself graphically thatgg (u)- u and assume that...
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