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5. (15 Points) Let T be a random variable that is the time to failure (in...
- (a) The failure time is 15 points) opns below PDF years (x) of a component...
- (a) The failure time is 15 points) opns below PDF years (x) of a component has the probabilsty density function ce o elsewer Find the probability that the component will fail in the first 2 years P( x S 2) (b) A system includes four components (A, B, and C), one of which will fail overa time period. The probabilities of the mutually exclusive component failures are P(C)-0.25 P(D) 0.10 P(A) 0.20 P(B) 0.15 The probability ofa system failure...
6. (15 points) Let X be an exponential random variable with mean 3. Answer the following:...
6. (15 points) Let X be an exponential random variable with mean 3. Answer the following: (a) Find the probability density function f(x). (b) Compute the probabilities P(2 < X < 6) and P(X 24). (c) Find the variance.
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter...
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
3. Let X be a continuous random variable with probability density function ax2 + bx f(0)...
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
Please help with this question. 12. (15 points) Let X be a continuous random variable with...
Please help with this question.
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0. F(x) = Inc. <a a<x<b bcx 1. (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
1. A device that continuously measures and records seismic activity is placed in a remote region. The time T, failure of the device is exponentially distributed with mean 3 years Since the device wil...
1. A device that continuously measures and records seismic activity is placed in a remote region. The time T, failure of the device is exponentially distributed with mean 3 years Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X max(T, 2). Then E(X) - 2. The loss due to fire in a commercial building is modelled by a random variable X with probability density function f(x)-(0.00020-x)...
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F()...
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a...
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a F(x) = Inr, asi<b 1, bsa (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(x > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0 < 1, where...
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0 < 1, where c is a constant. i) Find the constant c ii) What is the distribution function of X? ii) Let Y 1x<0.5 Find the conditional expectation E(X|Y).
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a...
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a Inz, a<<b 1, bsa (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function S(x) for X. (d) Find E(X)
- (a) The failure time is 15 points) opns below PDF years (x) of a component has the probabilsty density function ce o elsewer Find the probability that the component will fail in the first 2 years P( x S 2) (b) A system includes four components (A, B, and C), one of which will fail overa time period. The probabilities of the mutually exclusive component failures are P(C)-0.25 P(D) 0.10 P(A) 0.20 P(B) 0.15 The probability ofa system failure...
6. (15 points) Let X be an exponential random variable with mean 3. Answer the following: (a) Find the probability density function f(x). (b) Compute the probabilities P(2 < X < 6) and P(X 24). (c) Find the variance.
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
Please help with this question.
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0. F(x) = Inc. <a a<x<b bcx 1. (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
1. A device that continuously measures and records seismic activity is placed in a remote region. The time T, failure of the device is exponentially distributed with mean 3 years Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X max(T, 2). Then E(X) - 2. The loss due to fire in a commercial building is modelled by a random variable X with probability density function f(x)-(0.00020-x)...
12. (15 points) Let X be a continuous random variable with cumulative distribution function **- F() = 0, <a Inx, a < x <b 1, b<a (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a F(x) = Inr, asi<b 1, bsa (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(x > 2). (c) Find the probability density function f(x) for X. (d) Find E(X)
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0 < 1, where c is a constant. i) Find the constant c ii) What is the distribution function of X? ii) Let Y 1x<0.5 Find the conditional expectation E(X|Y).
1. (15 points) Let X be a continuous random variable with probability density function f (x) c(1-), 0
12. (15 points) Let X be a continuous random variable with cumulative distribution function 0, <a Inz, a<<b 1, bsa (a) Find the values of a and b so that F(x) is the distribution function of a continuous random variable. (b) Find P(X > 2). (c) Find the probability density function S(x) for X. (d) Find E(X)
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