Let X be a random variable that follows the exponential distribution with a rate parameter 1. (a) Compute P(X > 0.5|X > 0.3). (b) Show that the moment generating function of X is Mx(t) = iid Exponential(1). Use moment generating func- (c) Suppose that you have X1,., X30 tions to show that X; follows the Gamma distribution with shape parameter 30 and rate parameter 1. Hint: You may use the following fact without proving it: the mgf of Gamma-distribution with parameter a (shape) and 3 (rate) is (1- )-a. 25

MATLAB: An Introduction with Applications
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Let X be a random variable that follows the exponential distribution with a rate parameter
1.
(a) Compute P(X > 0.5[X > 0.3).
(b) Show that the moment generating function of X is Mx(t) =
iid
Exponential(1). Use moment generating func-
(c) Suppose that you have X1,.., X30
tions to show that Xi follows the Gamma distribution with shape parameter 30
and rate parameter 1.
Hint: You may use the following fact without proving it: the mgf of Gamma-distribution
with parameter a (shape) and 3 (rate) is (1 – )-a.
(d) Let X
P(X < 1.2)?
25
25 E Xi. How would you use the central limit theorem to approximate
i3D1
Transcribed Image Text:Let X be a random variable that follows the exponential distribution with a rate parameter 1. (a) Compute P(X > 0.5[X > 0.3). (b) Show that the moment generating function of X is Mx(t) = iid Exponential(1). Use moment generating func- (c) Suppose that you have X1,.., X30 tions to show that Xi follows the Gamma distribution with shape parameter 30 and rate parameter 1. Hint: You may use the following fact without proving it: the mgf of Gamma-distribution with parameter a (shape) and 3 (rate) is (1 – )-a. (d) Let X P(X < 1.2)? 25 25 E Xi. How would you use the central limit theorem to approximate i3D1
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