The lifetime of a machine is continuous on the interval (0, 40) with probability density function...
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The lifetime of a machine is continuous on the interval (0,
40) with probability density function...
1. Let X be a continuous random variable with probability density function f(x) = { if...
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
The probability density function of X, the lifetime of a certain type of electronic device (measured...
The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by fX(x) = ( C/x^2 x>5 0 x<5 where C>0 is a constant which needs to be determined. (i) What is the probability that the device’s lifetime is 10 hours? (ii) Find the 25%th quantile of X? (iii) If the device lifetime is X, then its total electricity cost equals . What is the expected total electricity cost of the...
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
Let X be a continuous random variable defined on the interval [0, 4] with probability density...
Let X be a continuous random variable defined on the interval [0, 4] with probability density function p(x) = c(1 + 4x) (a) Find the value of c such that p(x) is a valid probability density function. (b) Find the probability that X is greater than 3. (c) If X is greater than 1, find the probability X is greater than 2. (d) What is the probability that X is less than some number a, assuming 0 < a <...
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...
{ 2te-2 t> 0 6. Let g(t) be the probability density function of the continuous 0...
{ 2te-2 t> 0 6. Let g(t) be the probability density function of the continuous 0 t< 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X <m) = } = 0.5. [7]
be a continuous random variable with probability density function 3. Let for 0 r 1 a,...
be a continuous random variable with probability density function 3. Let for 0 r 1 a, for 2 < < 4 0, elsew here 2 7 fx(x) = (a) Find a to make fx(x) an acceptable probability density function. (b) Determine the (cumulative) distribution function F(x) and draw its graph.
6. Let X be a continuous random variable whose probability density function is: 0, x <0,...
6. Let X be a continuous random variable whose probability density function is: 0, x <0, x20.5 Find the median un the mode. 7. Let X be a continuous random variable whose cumulative distribution function is: F(x) = 0.1x, ja 0S$s10, Find 1) the densitv function of random variable U-12-X. 0, ja x<0, I, ja x>10.
7.2. Which of the following functions represent a probability density function for a continuous random variable?...
7.2. Which of the following functions represent a probability density function for a continuous random variable? Hint: Check if both rules of a proper probability density function hold. (a) f(z) = 0.25 where 0-1-8. b) f(r) =1/2 where 0 <1<2
Q 2. The probability density function of the continuous random variable X is given by Shell,...
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and 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
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...
{ 2te-2 t> 0 6. Let g(t) be the probability density function of the continuous 0 t< 0 random variable X. a. Verify that g(t) is indeed a probability density function. [8] b. Find the median of X, i.e. the number m such that P(X <m) = } = 0.5. [7]
be a continuous random variable with probability density function 3. Let for 0 r 1 a, for 2 < < 4 0, elsew here 2 7 fx(x) = (a) Find a to make fx(x) an acceptable probability density function. (b) Determine the (cumulative) distribution function F(x) and draw its graph.
6. Let X be a continuous random variable whose probability density function is: 0, x <0, x20.5 Find the median un the mode. 7. Let X be a continuous random variable whose cumulative distribution function is: F(x) = 0.1x, ja 0S$s10, Find 1) the densitv function of random variable U-12-X. 0, ja x<0, I, ja x>10.
7.2. Which of the following functions represent a probability density function for a continuous random variable? Hint: Check if both rules of a proper probability density function hold. (a) f(z) = 0.25 where 0-1-8. b) f(r) =1/2 where 0 <1<2
Q 2. The probability density function of the continuous random variable X is given by Shell, -<< 0. elsewhere. f(x) = {&e*, -40<3<20 (a) Derive the moment generating function of the continuous random variable X. (b) Use the moment generating function in (a) to find the mean and variance of X.
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