The lifetime, X, of a heavily used glass door has an exponential distribution with rate of =0.25 per year. Thus, the density of X is:

f(x,) = e^x

for 0 x , = 0.25 . is what R calls rate. Hint: This is a problem involving the exponential distribution. Knowing the parameter for the distribution allows you to easily answer parts a ,b ,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts . Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts. a) What is the expected value of X? b) What is the variance of X? c) What is the standard deviation of X? d) What is the probability that X is greater than its expected value? e) What is the probability that X is > 5? f) What is the probability that X is > 10? g) What is the probability that X > 10 given that X > 5? h) What is the median of X?

question 02

The mean annual loss from fire for homeowners in New York State is = $220 with a standard deviation of = $140. Suppose we randomly select 144 New York homeowners and measure the fire loss for each homeowner. Let M be the random variable representing the mean fire loss of the 144 selected homeowners. Let T be the random variable representing the total amount (sum) of the fire losses for the 144 selected homeowners a) What theorem will let us treat T and M as approximately normal random variables?Law of Large Numbers301 Theorem Monte Carlo TheoremConvolution TheoremCentral Limit TheoremChebychev's Theorem b) What is the expected value of T? c) What is the standard deviation of T? d) What is the approximate probability that T is greater than 30000? e) What is the 90th percentile of the approximate distribution of T? f) What is the expected value of M? g) What is the standard deviation of M? h) What is the approximate probability M is greater than 225?

question 03

1. 1. The amount, X, that Jim Jack overestimates his game playing reflex proficiency has the following probability density function: f(x)=

x²2

57

for

3 x 6

and 0 otherwise. a) What is the probability that X > 4? b) What is the probability that X < 4? c) What is the probability that 3.1 < X < 4.7? d) What is the expected value of X (E(X))? e) What is the expected value of X - 2? f) What is the expected value of 5X? g) What is the expected value of

X²

? h) What is the probability that X is greater than its expected value? i) What is the expected value of

X³2X1

? j) What is the 60th percentile of X? k) What is the probability that X is within 0.3 of its expected value? l) What is the probability that X = 4?