Imp. Distributions

1

We must find the difference between the two probabilities

P(Y=y) = P(Y≤y) - P(Y≤y-1) = [(k-y+1)/k]^n - [(k-y)/k]^n

Thus: P(Y=y) = [(k-y+1)^n - (k-y)^n]/k^n

2a

cat("Prob not failing 8 yrs geometric=", (9/10)^8, "\n",
    "Expected=", 10, "\n",
    "SD",sqrt((1-1/10)/(1/10)^2) )

## Prob not failing 8 yrs geometric= 0.4304672 
##  Expected= 10 
##  SD 9.486833

2b

cat("Prob not failing 8 yrs exponential=", exp(-8/10), "\n",
    "Expected=", 10, "\n",
    "SD",10) 

## Prob not failing 8 yrs exponential= 0.449329 
##  Expected= 10 
##  SD 10

2c

cat("Prob not failing 8 yrs exponential=", dbinom(0,8,0.1), "\n",
    "Expected=",0.1*8 , "\n",
    "SD",sqrt(0.8*0.9)) 

## Prob not failing 8 yrs exponential= 0.4304672 
##  Expected= 0.8 
##  SD 0.8485281

#2d

cat("Prob not failing 8 yrs exponential=", ppois(0,lambda = 0.8), "\n",
    "Expected=", 0.8, "\n",
    "SD",sqrt(8/10)) 

## Prob not failing 8 yrs exponential= 0.449329 
##  Expected= 0.8 
##  SD 0.8944272