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A random variable X obeys the exponential distribution law xp-x),(20). m! Prove that the following inequality...
X is a random variable with exponential distribution whose expectation is . Prove : We were...
X is a random variable with exponential distribution whose expectation is . Prove : We were unable to transcribe this imageた! E(Xk) =E, k = 1.2. 3
6. The exponential distribution Consider the random variable x that follows an exponential distribution, with p...
6. The exponential distribution Consider the random variable x that follows an exponential distribution, with p - 10. The standard deviation of X is o = The parameter of the exponential distribution of X is A - What is the probability that X is less than 7? OP(X < 7) = 0.3935 OP(X < 7) = 0.5034 OP(X < 7) = 0.4908 OPIX < 7) = 7981 What is the probability that X is between 12 and 20? O P...
exponential distribution 3. The distribution of Smith's future lifetime is X, an exponential random variable with...
exponential distribution
3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution of Brown's future lifetime is Y, an exponential random variable with mean B. Smith and Brown have future lifetimes that are independent of one another. Find the probability that Smith outlives Brown. Answer #3: (D) a (E) (A) (B) (C)
3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution...
Problem 1.33. Let X be an exponential random variable with unit rate Fix two positive numbers...
Problem 1.33. Let X be an exponential random variable with unit rate Fix two positive numbers x and y. Prove that P(X > x+91X > x) P(X > y). This shows that conditioning the exponential clock on not having rung by time r and then restarting the count at that point gives statistically the same exponential clock! This is called the memoriless property of the exponential distribution. The same holds for the geometric distribution.
chebyshev’s inequality Problem 2 Chebyshev's Inequality Suppose that the random variable ? has a Poisson distribution...
chebyshev’s inequality
Problem 2 Chebyshev's Inequality Suppose that the random variable ? has a Poisson distribution with the parameter ? > 0, ~ ?(A). Using the Chebyshev's inequality prove that Problem 3 - Application of the Chebyshev's Inequality Suppose that a player plays a game where he gains a dollar with the probability or loses a dollar with the probability . That is, his gain from one game can be modeled as a random variable fi, such that If the...
Prove that for positive real numbers x and y, the following inequality holds:
Prove that for positive real numbers x and y, the following inequality holds:
4. Let X be a Exponential random variable, X ~ Expo(2). Find the pdf of X3....
4. Let X be a Exponential random variable, X ~ Expo(2). Find the pdf of X3. [Hint: pdf of XP is not (pdf of X)3, find it by differentiating the cdf of X3, i.e., Px() = P(X® S2)]
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with ...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF
Problem 5. Let X...
Problem #3. X is a random variable with an exponential distribution with rate 1 = 3...
Problem #3. X is a random variable with an exponential distribution with rate 1 = 3 Thus the pdf of X is f(x) = le-ix for 0 < x where = 3. a) Using the f(x) above and the R integrate function calculate the expected value of X. b) Using the dexp function and the R integrate command calculate the expected value of X. c) Using the pexp function find the probability that .4 SX 5.7 d) Calculate the probability...
6. The exponential distribution Consider the random variable x that follows an exponential distribution, with p - 10. The standard deviation of X is o = The parameter of the exponential distribution of X is A - What is the probability that X is less than 7? OP(X < 7) = 0.3935 OP(X < 7) = 0.5034 OP(X < 7) = 0.4908 OPIX < 7) = 7981 What is the probability that X is between 12 and 20? O P...
exponential distribution
3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution of Brown's future lifetime is Y, an exponential random variable with mean B. Smith and Brown have future lifetimes that are independent of one another. Find the probability that Smith outlives Brown. Answer #3: (D) a (E) (A) (B) (C)
3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution...
Problem 1.33. Let X be an exponential random variable with unit rate Fix two positive numbers x and y. Prove that P(X > x+91X > x) P(X > y). This shows that conditioning the exponential clock on not having rung by time r and then restarting the count at that point gives statistically the same exponential clock! This is called the memoriless property of the exponential distribution. The same holds for the geometric distribution.
chebyshev’s inequality
Problem 2 Chebyshev's Inequality Suppose that the random variable ? has a Poisson distribution with the parameter ? > 0, ~ ?(A). Using the Chebyshev's inequality prove that Problem 3 - Application of the Chebyshev's Inequality Suppose that a player plays a game where he gains a dollar with the probability or loses a dollar with the probability . That is, his gain from one game can be modeled as a random variable fi, such that If the...
4. Let X be a Exponential random variable, X ~ Expo(2). Find the pdf of X3. [Hint: pdf of XP is not (pdf of X)3, find it by differentiating the cdf of X3, i.e., Px() = P(X® S2)]
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF
Problem 5. Let X...
Problem #3. X is a random variable with an exponential distribution with rate 1 = 3 Thus the pdf of X is f(x) = le-ix for 0 < x where = 3. a) Using the f(x) above and the R integrate function calculate the expected value of X. b) Using the dexp function and the R integrate command calculate the expected value of X. c) Using the pexp function find the probability that .4 SX 5.7 d) Calculate the probability...
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