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2. Let X be an exponential random variable with rate A > 0. In this problem...
(c) Let X have an exponential density with parameter λ > 0, Prove the "memoryless" property:...
(c) Let X have an exponential density with parameter λ > 0, Prove the "memoryless" property: P(X > t + s|X > s) = P(X > t) for t>0 and s 0. For example, the probability that the conversation lasts at least t more minutes is the same as the probability of it lasting at least t minutes in the first place.
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...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
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.
3. Let X be a (discretel) random variable having the following pmf: P(X= k) = ....
3. Let X be a (discretel) random variable having the following pmf: P(X= k) = . k=0, 1, .... Check that this is indeed a pmf. Show that for any non-negative integers a <b we have P(X>b|X > a) = P(X> b-a) words, X satisfies the same sort of "memoryless" property as the Exponential distribution.
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s...
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
Let X be the random variable with the geometric distribution with parameter 0 < p <...
Let X be the random variable with the geometric distribution with parameter 0 < p < 1. (1) For any integer n ≥ 0, find P(X > n). (2) Show that for any integers m ≥ 0 and n ≥ 0, P(X > n + m|X > m) = P(X > n) (This is called memoryless property since this conditional probability does not depend on m.)
Problem 4. Let X be a random variable with EIXI4 < oo. Define μ1 = EX...
Problem 4. Let X be a random variable with EIXI4 < oo. Define μ1 = EX and Alk-E(X-μ)k, k 2, 3, 4, and then 03 = 쓺 (skewness), a,--2 (kurtosis) 3/2 (1) Show that if P(X- > z) = P(X-円く-r) for every x > 0, then μ3-0, but not the other way around. (2) Compute as and a when X is Binomial with parameter p, exponential with mean1, uniform on [O, 1], standard normal, and double exponential (fx (x)-(1/2)e-M).
Let X be an exponential random variable such that P(X < 27) = P(X > 27)....
Let X be an exponential random variable such that P(X < 27) = P(X > 27). Calculate E[X|X > 23].
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an...
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an exponential distribution, written Exp(X), if its probability density function f is such that le- if > 0 10 if x < 0 f(0) = 0 where > 0 is a real number. 1. Compute the mean of X 2. Compute the variance of X 3. Compute the cumulative distribution function F of X. Use this to show that for any real numbers s and...
(c) Let X have an exponential density with parameter λ > 0, Prove the "memoryless" property: P(X > t + s|X > s) = P(X > t) for t>0 and s 0. For example, the probability that the conversation lasts at least t more minutes is the same as the probability of it lasting at least t minutes in the first place.
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...
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
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.
3. Let X be a (discretel) random variable having the following pmf: P(X= k) = . k=0, 1, .... Check that this is indeed a pmf. Show that for any non-negative integers a <b we have P(X>b|X > a) = P(X> b-a) words, X satisfies the same sort of "memoryless" property as the Exponential distribution.
3. Let X be an exponential random variable with parameter 1 = $ > 0, (s is a constant) and let y be an exponential random variable with parameter 1 = X. (a) Give the conditional probability density function of Y given X = x. (b) Determine ElYX]. (c) Find the probability density function of Y.
Problem 4. Let X be a random variable with EIXI4 < oo. Define μ1 = EX and Alk-E(X-μ)k, k 2, 3, 4, and then 03 = 쓺 (skewness), a,--2 (kurtosis) 3/2 (1) Show that if P(X- > z) = P(X-円く-r) for every x > 0, then μ3-0, but not the other way around. (2) Compute as and a when X is Binomial with parameter p, exponential with mean1, uniform on [O, 1], standard normal, and double exponential (fx (x)-(1/2)e-M).
Let X be an exponential random variable such that P(X < 27) = P(X > 27). Calculate E[X|X > 23].
Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an exponential distribution, written Exp(X), if its probability density function f is such that le- if > 0 10 if x < 0 f(0) = 0 where > 0 is a real number. 1. Compute the mean of X 2. Compute the variance of X 3. Compute the cumulative distribution function F of X. Use this to show that for any real numbers s and...
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