Lab 3-Peter Tzanetos

Lab 3 2023-02-13 Name:Peter Tzanetos Section: CX07 1A rbinom( 10 , 1 , 0.2 ) ## [1] 0 0 0 1 0 0 0 0 0 0 The relative frequency of heads in this test was 3/10. For 10 Heads 0 Tails Probability=0.0000001024 For 9 Heads 1 Tail Probability=0.0000004096 For 8 Heads 2 Tails Probability=0.000002048 For 7 Heads 3 Tails Probability= 0.0000065536 For 6 Heads 4 Tails Probability=0.0000262144 For 5 Heads 5 Tails Probability=0.0001048576 For 4 Heads 6 Tails Probability=0.0004194304 For 3 Heads 7 Tails Probability=0.0016777216 For 2 Heads 8 Tails Probability=0.0067108864 For 1 Heads 9 Tails Probability=0.0268435456 For 0 Heads 10 Tails Probability=0.1073741824 1B. coin .10000 <- sample(c( 0 , 1 ), 10000 , replace = T, prob = c( 0.8 , 0.2 )) heads .10000 <- cumsum(coin .10000 == 1 ) # sequentially record the number of heads heads.prop <- heads .10000 /( 1 : 10000 ) # record the proportion of heads plot( 1 : 10000 , heads.prop, type= "l" , xlab= "Number of Flips" , ylab= "Frequency o f Heads" ) lines( 1 : 10000 , rep( 0.2 , 10000 ), col= "red" )

2A. The probability of P(X > 2) and P(X ≤ 4) is 20 percent. 2B. The expected value is 0 0.3+2 0.2+4 0.2+6 .03=3 2C.The Variance is 6.667 and the Standard Deviation is 2.58 2D. X <- sample(c( 0 , 2 , 4 , 6 ), 10000 , replace= T, prob = c( 0.3 , 0.2 , 0.2 , 0.3 )) table(X)/ 10000 # calculating the observed frequencies ## X ## 0 2 4 6 ## 0.3037 0.1984 0.2023 0.2956 sum(X> 2 )/ 10000 # observed frequency of X less than 3 ## [1] 0.4979 sum(X<= 4 )/ 10000 # observed frequency of X less than 3 ## [1] 0.7044 sum(X<= 4 )/ 10000 -sum(X> 2 )/ 10000 ## [1] 0.2065 The probability in the simulation came out to 19.59% which is very close to my predicted 20 percent. mean(X) ## [1] 2.9796

var(X) ## [1] 5.794563 The expected value came out extremely close to my projection of 3 and the variance was close to my projection of 6.667, but a little bit off. This was the most likely to be farther off as variance is varies the most of the three. 2E. Y <- table(X) # create table of discrete random variable X names(Y) <- c( "0" , "2" , "4" , "6" ) # assign label names barplot(Y, main= "Bar Chart of 10000 Samples" , col= c( "purple" , "orange" , "green" , "blue" ), xlab= "Category" , ylab= "Frequency" ) # create bar chart 3A. Y <- rbinom( 20000 , 8 , 0.5 ) sum(Y== 5 )/ 20000 ## [1] 0.22095 The chances of it equaling 5 are 22.04%. 3B. sum(Y>= 4 )/ 20000 ## [1] 0.63885 sum(Y< 7 )/ 20000 -sum(Y>= 4 )/ 20000 ## [1] 0.3267

The chances of it being above 4 and below 7 are 32.93%. X <- rbinom( 2000 , 8 , 0.5 ) barplot(table(X), col= c( "yellow" , "red" , "green" , "purple" , "orange" ), xlab= "Catego ry" , ylab= "Frequency" , main= "Bar Chart of 20000 Samples" )