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TOPIC:Drawing random sample from Erlang distribution.
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Suppose that X is an exponential randon variable with λ- 3 and we know how to...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
Suppose X has an exponential distribution with parameter λ = 1 and Y |X = x...
Suppose X has an exponential distribution with parameter λ = 1 and Y |X = x has a Poisson distribution with parameter x. Generate at least 1000 random samples from the marginal distribution of Y and make a probability histogram.
3. If X follows an exponential distribution with mean 1/λ. Find the density function of Y,...
3. If X follows an exponential distribution with mean 1/λ. Find the density function of Y, where (b) Y = 1/x.
Recall that X ∼ Exp(λ) if the probability density function of X is fX(x) = λe−λx...
Recall that X ∼ Exp(λ) if the probability density function of X
is fX(x) = λe−λx for x ≥ 0. Let X1, . . . , Xn ∼ Exp(λ), where λ is
an unknown parameter. Exponential random variables are often used
to model the time between rare events, in which case λ is
interpreted as the average number of events occurring per unit of
time.
Recall that X ~ Exp(A) if the probability density function of X is fx(x)-Ae-Az for...
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in h...
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have...
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
4. Suppose that X and Y are independent and follow an exponential distri- bution with parameter...
4. Suppose that X and Y are independent and follow an exponential distri- bution with parameter A. Show that the random variable Z min X,Y also follows an exponential distribution, with parameter 2λ. (hint: we have min(X, Y\ 2 z if and only if X 2 z and Y2 2)
3. If X is an exponential random variable with parameter λ > 0, show that for...
3. If X is an exponential random variable with parameter λ > 0, show that for c > 0 cX is exponential with parameter λ/c.
2) with parameter λ = 2. Now, we have a unbiased coin. We throw it. If we get tail, we take the n...
2) with parameter λ = 2. Now, we have a unbiased coin. We throw it. If we get tail, we take the number X. If we get head we take 3 times X. The result is called Z. What is the probability density of Z. (Read up about the probability density of exponential variable online). So, in other words, we generate a random number X which is exponential with parameter λ 2, then, we flip a coin. When we get...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
1. Suppose that Xi,..,Xn are independent Exponential random variables with density f(x; λ) λ exp(-1x) for x > 0 where λ > 0 is an unknown parameter (a) Show that the τ quantile of the Exponential distribution is F-1 (r)--X1 In(1-7) and give an approximation to Var(X(k)) for k/n-T. What happens to this variance as τ moves from 0 to 1? (b) The form of the quantile function in part (a) can be used to give a quantile-quantile (QQ) plot...
3. If X follows an exponential distribution with mean 1/λ. Find the density function of Y, where (b) Y = 1/x.
Recall that X ∼ Exp(λ) if the probability density function of X
is fX(x) = λe−λx for x ≥ 0. Let X1, . . . , Xn ∼ Exp(λ), where λ is
an unknown parameter. Exponential random variables are often used
to model the time between rare events, in which case λ is
interpreted as the average number of events occurring per unit of
time.
Recall that X ~ Exp(A) if the probability density function of X is fx(x)-Ae-Az for...
1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in humans follows a Poisson distribution whose parameter depends on the observed individual. This means that for Jason we have X ~ Poi(mj), where mj is Jason's parameter value, while for Alice we have X ~Poi(mA), where mA is Alice's parameter value. For a person selected at random we may consider the parameter value M as a random variable such that, given that M, we have...
4. Suppose that X and Y are independent and follow an exponential distri- bution with parameter A. Show that the random variable Z min X,Y also follows an exponential distribution, with parameter 2λ. (hint: we have min(X, Y\ 2 z if and only if X 2 z and Y2 2)
2) with parameter λ = 2. Now, we have a unbiased coin. We throw it. If we get tail, we take the number X. If we get head we take 3 times X. The result is called Z. What is the probability density of Z. (Read up about the probability density of exponential variable online). So, in other words, we generate a random number X which is exponential with parameter λ 2, then, we flip a coin. When we get...
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