1. Two efficiency experts take independent measurements Y₁ and Y₂ on the length of time workers take to complete a certain task. Each measurement is assumed to have the probability density function given by fy(y) = favoz. 0, y > 0 otherwise (a) Find the moment generating function of Y₁. Clearly show your steps. Y₁+Y₂ 2 Clearly show your steps. (b) Find the probability density function for the average X = (c) Find E(X) and V(X). Justify your answers. (d) Suppose that X₁, X2, ....X100 are 100 independent observations from a probability distribution of X and X is the sample mean of the random sample. How can the value xo such that P(X < xo) = 0.90 be found in R? Select the appropriate R function from those listed below, and write down the R code, with the arguments filled in appropriately, that would give the value xo such that P(X < xo) = = 0.90. Note that you do not need to report a numerical answer here. Some R functions are included here for your reference: pexp (a, rate), qexp (a, rate), dexp (a, rate), pnorm (a, mean, sd), qnorm (a, mean, sd), dnorm (a, mean, sd), punif (a, min, max), qunif (a, min, max), dunif (a, min, max), qgamma (a, alpha, rate), dgamma (a, alpha, rate) pgamma (a, alpha, rate),

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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Chapter10: Statistics
Section10.1: Measures Of Center
Problem 9PPS
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1. Two efficiency experts take independent measurements Y₁ and Y₂ on the length of time workers take
to complete a certain task. Each measurement is assumed to have the probability density function given
by
fy(y) =
-2/27,
0,
y>0
otherwise
(a) Find the moment generating function of Y₁. Clearly show your steps.
(b) Find the probability density function for the average X =
(c) Find E(X) and V(X). Justify your answers.
(d) Suppose that Xx₁, x2,....X100 are 100 independent observations from a probability distribution of
X and X is the sample mean of the random sample. How can the value xo such that P(X < xo)
= 0.90 be found in R? Select the appropriate R function from those listed below, and write down
the R code, with the arguments filled in appropriately, that would give the value xo such that
P(X<xo) = 0.90. Note that you do not need to report a numerical answer here. Some R
functions are included here for your reference:
pexp (a, rate), qexp (a, rate), dexp (a, rate), pnorm (a, mean, sd),
qnorm (a, mean, sd), dnorm (a, mean, sd), punif (a, min, max),
qunif (a, min, max), dunif (a, min, max), pgamma (a, alpha, rate),
qgamma (a, alpha, rate), dgamma (a, alpha, rate)
Y₁+%2. Clearly show your steps.
2
Transcribed Image Text:1. Two efficiency experts take independent measurements Y₁ and Y₂ on the length of time workers take to complete a certain task. Each measurement is assumed to have the probability density function given by fy(y) = -2/27, 0, y>0 otherwise (a) Find the moment generating function of Y₁. Clearly show your steps. (b) Find the probability density function for the average X = (c) Find E(X) and V(X). Justify your answers. (d) Suppose that Xx₁, x2,....X100 are 100 independent observations from a probability distribution of X and X is the sample mean of the random sample. How can the value xo such that P(X < xo) = 0.90 be found in R? Select the appropriate R function from those listed below, and write down the R code, with the arguments filled in appropriately, that would give the value xo such that P(X<xo) = 0.90. Note that you do not need to report a numerical answer here. Some R functions are included here for your reference: pexp (a, rate), qexp (a, rate), dexp (a, rate), pnorm (a, mean, sd), qnorm (a, mean, sd), dnorm (a, mean, sd), punif (a, min, max), qunif (a, min, max), dunif (a, min, max), pgamma (a, alpha, rate), qgamma (a, alpha, rate), dgamma (a, alpha, rate) Y₁+%2. Clearly show your steps. 2
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