Question

I must use R Program to solve them. Please help! Thank you

ünif uniform random variable 1) Draw the graphs of the p.d.f. of the following distributions (a) The standard normal p.d.f (b) The normal pdf with ? = 50, ? = 10 (c) The uniform p.d.f. over interval [10, 20] (d) The exponential P.d.f with parameter ? 4. 2) Illustrating the central limit theorem. Let X be a random variable having the uniform distribution over the interval [6, 12] Denote by X1, X. X3, a sequence of independent random variables with the same distribution as X. Define the sample mean by X The central limit theorem applied to this particular case implies that the probability distribution of 7 converges to the standard normal distribution for certain values of ? and ? (a) For what values of and ? does the convergence hold? (This is to be done by hand.) (b) For each of the four values n-2, 5, 10, and 20 do the following: Obtain 10000 independent values of the sample mean random variable X, where the X are generated from the uniform distribution over the interval [6, 12]; then create a histogram of for eachn. Discuss what you notice for each case on n.

Answer 1

1)

a) The graph of $N\left ( 0,1 \right )$ is drawn below using R.

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R code given below.

x <- seq(-5, 5, length=1000)
y <- dnorm(x, mean=0, sd=1)
plot(x, y, type="l", lwd=1)

b)The graph of $N\left ( 50,10^2 \right )$ is drawn below using R.

R code given below.

0.03 0.04 0.01 0.02 0.00 3 8

x <- seq(10, 90, length=1000)
y <- dnorm(x, mean=50, sd=10)
plot(x, y, type="l", lwd=1)

c) The graph of $Unif\left ( 10,20 \right )$ is drawn below using R.

0.00 0.02 0.04 0.06 0.08 0.10 8

R code given below.

x <- seq(5, 25, length=1000)
y <- dunif(x, 10, 20)
plot(x, y, type="l", lwd=1)

d) The graph of $Exp\left ( \lambda =4\right )$ is drawn below using R.

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R code given below.

x <- seq(5, 10, length=1000)
y <- dexp(x, 4)
plot(x, y, type="l", lwd=1)