Question

R studio

#Exercise : Calculate the following probabilities :

#1. Probability that a normal random variable with mean 22 and variance 25
#(i)lies between 16.2 and 27.5

#(ii) is greater than 29

#(iii) is less than 17

#(iv)is less than 15 or greater than 25

#2.Probability that in 60 tosses of a fair coin the head comes up
#(i) 20,25 or 30 times

#(ii) less than 20 times

#(iii) between 20 and 30 times

#3.A random variable X has Poisson distribution with mean 7. Find the probability that

# (i) X is less than or equal to 5:

#less than 4:

#(ii) X is greater than 10

# (iii)X is between 4 and 16

#Quantiles
# The following examples show how obtain the quantiles
# of some common distributions for the values (vector) given:

#Normal(0,1) Distribution :

y <- c(.01,.05,.1,.2,.5,.8,.95,0.975, .99)

#BinomialDistribution size = 30, prob = 0.2:

y <- c(.01,.05,.1,.2,.5,.8,.95,.99)

#Poisson($\lambda$) Distribution with mean (lambda) = 6:

y <- c(.01,.05,.1,.2,.5,.8,.95,.99)

#Random Variable generation

# Generate 10 random numbers from the normal (0,1) distribution

# Generate 10 random numbers from the normal distribution with mean 5 and a standard deviation of 2.5

# Generate 20 random numbers from the binomial with a size of 5 and probability of 20 percent
# Generate 20 random numbers from the poisson with a lambda of 6

# Exercise (Advanced) : Generate 500 samples from Student's $t$ distribution with 5 degrees of freedom and plot the historgam.
# (Note: $t$ distribution is going to be covered in class). The corresponding function is rt . hist(rt(500,5),40)
#Density Plots

#Plotting the probability density function (pdf) of a Normal distribution :
# over the x vector given; and plot it

x <- seq(-4.5,4.5,.1)

#Plotting the probablity mass function for the following 2 variables distributed Binomially with size 30;
# the first one with probability 0.15 and the second with probability 0.4:
# Split the screen into 2 and generate the two plots on the same screen
k <- c(1:15)

#================================================================================================

#Using the dataset "USAarrests" available in base R; answer the questions below.
# This data set contains statistics about violent crime rates by us state; it is available in base R

#Make a scatterplot of Murder and UrbanPop. Describe the relationship between these two variables.

# display histograms of Murder and Rape

# Find the mean and standard deviation of both Murder and Rape

# Identify the 27th percentile of the Assault series; the 50th percentile

# # Identify the 27th percentile of the Rape data, the 50th percentile AND the 97the percentile

# display a barchart of the 27th, 50th, and 97th percentiles

# Determine what proportion of the data
# are within one standard deviation of the mean

# Display a histogram of Assault and Rape and identify the mean and the mean +/- one standard deviation

# Provide a title and properly label the x and y axes for all your charts.

#========================================================================================================

# Compare monthly data for the "market" (eg DJI) and for General Electric for the period 2015-01 till September
# #Rebase both sector series to 100 in 2015-01-01
# How was GE performed relative to the market over the period examined?
# Do this by rebasing the two series to 100 = 2015-01-01
# add an appropriate legend to the graph
# Color one line purple and the other red

#========================================================================================================

#===============================================================================================
# Display the distribution of the mean of the data for Size,below (hint: use a for loop)
# this means that we are using the collected sample (shown below) to simulate the p

# Example Widgets
# Generate a sequence of number from 1 to 20; call it Count
# Generate a sequence of 20 numbers from 200.5 to 355.125; add to it numbers from a random normal of
# mean 5 and standard deviation 10; call it Size
# first find the mean of Size
# Identify the mean in the histogram of the distribution of the mean - with a red line; and label the graph properly

#display the generated data for Size in a boxplot

# What is the probability that someone is less than 270 in size?
#5 What is the probability that a man is greater than 300 in size?
#6 What is the probaiblity of being between 280 and 300 in size?

# If being at 20% of the scale is the cutting point for being hefty when it comes to size, what is the "hefty cut-off point"?

# load library e1071; find measures of skewness and kurtosis of the Size variable.

Answer 1

Solution:-

1)

Mean = 22, variance = 25

(i) The probability that mean lies between 16.2 and 27.5 is 0.741.

x₁ = 16.2

x₂ = 27.5

By applying normal distribution:-

$z = \frac{x-\mu }{\sigma }$

z₁ = - 1.16

z₂ = 1.10

P( -1.16 < z < 1.10) = P(z > - 1.16) - P(z > 1.10)  

P( -1.16 < z < 1.10) = 0.877 - 0.136

P( -1.16 < z < 1.10) = 0.74

(ii) P(x > 29) = 0.081.

x = 29

By applying normal distribution:-

$z = \frac{x-\mu }{\sigma }$

z = 1.40

P(z > 1.40) = 0.081

(iii) P(x < 17) = 0.159

x = 17

By applying normal distribution:-

$z = \frac{x-\mu }{\sigma }$

z = - 1.0

P(z < -1.0) = 0.159

iv) P(15 > x > 25) = 0.355.

x₁ = 15

x₂ = 25

By applying normal distribution:-

$z = \frac{x-\mu }{\sigma }$

z₁ = - 1.40

z₂ = 0.60

P( -1.40 > z > 0.60) = P(z < -1.40) + P(z > 0.60)  

P( -1.40 > z > 0.60) = 0.081 + 0.274

P( -1.40 > z > 0.60) =  0.355