Question

Answer 5 and 6, using R programming code: 5. We have seen the following functions dnorm pnorm qnorm normal probability density function normal cumulative distribution function normal quantile function a. Let X have a normal distribution with mean 100 and variance 100. Find the 90th percentile of X by calling the function qnorm in two ways: (i) specify the arguments by position, and (ii) specify the arguments by complete names. b. Find P(X > 90) using the function pnorm in two ways: (i) without the use of the optional argument lower.tail, and (ii) with the use of the optional argument lower.tail but no additional arithmetic operation. 6. Let X have a Poisson distribution with parameter 1 = 3, the name for the Poisson distribution on R is pois, using one function per point find a. P(X = 4) b. P(X < 3) c. P(X > 2)

Answer 1

Answer Given Mean =100 variance=100 In our solution functions, Ruses standard deviation instead of variance, so we need to take the square root of variance to get standard deviation ! . we know that square of standard deviation is the variance Here variance is given as 100 in our problem. Hence standard deviation will be N100 =10

Code:

##Arguments by position##
##0.9 FOR 90th percentile, mean is 100
##and standard deviation is 10
qnorm(0.9, 100, 10)
##Arguments by names##
##0.9 FOR 90th percentile, mean is 100
##and standard deviation is 10
qnorm(0.9, mean = 100, sd = 10)
##90th percentile means P(X>90)
##By default pnorm gives the value of P(X<90)
##We can't change the tail values in this case
##So to use the value of P(X>90) we subtract the default value from 1
1-pnorm(0.9, mean = 100, sd = 10)
##90th percentile means P(X>90)
##By derault lower.tail=TRUE gives P(X<90)
##So to use the value of P(X>90) we can use lower.tail=FALSE
pnorm(0.9, mean = 100, sd = 10,lower.tail = FALSE)
##Use ppois function for pois distribution
##For X=4, we use the default value of lambda
##and need not use value of tail, so lower value is assigned to FALSE
ppois(4, lambda=3, lower=FALSE)
##For X<=3, we use the default value of lambda
##and as X is <=3, we can use default value of tail which is TRUE
ppois(3, lambda=3,lower.tail = TRUE)
##For X>2, we use the default value of lambda
##and as X is >2, we have to use the different value of tail which is FALSE
ppois(2, lambda=3,lower.tail = FALSE)