Suppose T₁ and T₂ are iid Exp(1).

What is the probability density function of T₁+T₂?

What is the probability that T₁+T₂ ≥ 3?

1. Given a population in where the probability of someone being left handed is 0.3,
people is taken then...

Calculate the probability the proportion of left-h

Find the​ z-score such that the interval within z standard deviations of the mean for a normal distribution contains

a. 34% of the probability

b. 94% of the probability

***IN JAVA***

Write a program contained a class Student which has firstName, lastName, mark, grade. The program should allow creation an array of object instances to assign firstName, lastName and mark from input user and perform and assign grade based on mark’s criteria displayed below.

SAMPLE OF OUTPUT:

Enter section #: 1

How many students: 3

Enter student details:

First Name: John

Last Name: Kendall

Marks      : 96

Enter student details:

First Name: Mary

Last Name: Kendall

Marks      : 76

Enter student details:

First Name: Lucy

Last Name: Kendall

Marks      : 76

Result for Section 1

-------------------------------------------------------------------

FirstName               Last Name         Grade

John                    Kendall           A+

Mary                    Kendall           C+

Lucy                    Kendall           C+

Total students: 3

Average marks: 82.2     Average Grade: B

Highest: 96, by John Kendall

Location: 0

Another batch? (Yes – 1, No - 0) :> 1

Enter section #: 2

How many students: 6

Enter student details:

1)

First Name: John

Last Name: Kendall

Marks      : 96

2)

First Name: Mary

Last Name: Kendall

Marks      : 76

3)

First Name: Lucy

Last Name: Kendall

Marks      : 76

4)

First Name: Martin

Last Name: Kendall

Marks      : 86

5)

First Name: Mary

Last Name: Kendall

Marks      : 76

6)

First Name: Lucy

Last Name: Kendall

Marks      : 98

Result for Section 2

-------------------------------------------------------------------

FirstName               Last Name         Grade

John                    Kendall           A+

Mary                    Kendall           C+

Lucy                    Kendall           C+

Martin                  Kendall           B+

Mary                    Kendall           C+

Lucy                    Kendall           A+

Total students: 6

Average marks : 80.7    Average Grade: B

Highest: 98, by Lucy Kendall

Location: 6

Another batch? (Yes – 1, No - 0) :> 0

PROBLEM SOLVING TIPS:

In this program, you will generate a random “sentence” constructed of random “words”. Quotes are used in the preceding sentence because the “words” will mostly be nonsense words that do not exist in the English language.

Step 1. The average number of words in an English sentence is about 17 words. First generate a pseudorandom number NW between 10 and 20 for the number of words in your random “sentence”. Use srand( ) to set the initial value in the iterative algorithm within the rand( ) function. Given NW, initialize a for-loop for(i=0; i

Step 2. To generate a pseudorandom “word” within the for-loop above, you first need to generate a pseudorandom number NL for the number of letters in the “word”.
The number of letters in a word follows the following probability table. two-letter words 0.20 three-letter words 0.27 four-letter words 0.22 five-letter words 0.14 six-letter words 0.09 seven-letter words 0.08 -------------------------------- Total = 1.00
Think of generating a pseudorandom value NL between 0.0 and 1.0 and choosing the number of letters in the “word” using the pseudorandom value according to the following: If NL >= (0.00) and NL <= (0.20), then the number of letters in the word will be two. If NL> (0.20) and NL <= (0.20+0.27), then the number of letters in the word will be three. If NL> (0.20+0.27) and NL <= (0.20+0.27+0.22), then the number of letters in the word will be four. If NL> (0.20+0.27+0.22) and NL <= (0.20+0.27+0.22+0.14), then the number of letters in the word will be five. If NL> (0.20+0.27+0.22+0.14) and NL <= (0.20+0.27+0.22+0.14+0.09), then the number of letters in the word will be six.
If NL> (0.20+0.27+0.22+0.14+0.09) and NL <= (0.20+0.27+0.22+0.14+0.09+0.08), then the number of letters in the word will be seven.
Notice that there are 6 intervals, representing NL=2 to NL=7. To assign a number of letters NL in the “word”, declare a 6-cell 1-D float array prob_letters[ ] and load the 1-D array with the probability sums from above. float prob_interval[6]={ 0.2, 0.47, 0.69, 0.83, 0.92, 1.0};
Now generate a pseudorandom value x between 0.0 and 1.0 x=rand( )/(float)RAND_MAX; Initialize the number of letters NL to 2: NL=2; Now check if x lies in one of the other five intervals for NL=3 to NL=7. If so, assign NL to the number of letters for that interval. for(i=1; i<6; i++){ if(x>= prob_interval[i-1] && x<= prob_interval[i])NL=i+2; }
Two important Notes : Important Note 1 : A better way to generate the 1-D array of interval values prob_interval[ ] is to start from the probabilities for the occurrencess of NL=2 through NL=7. float prob_interval[6]={ 0.2, 0.27, 0.22, 0.14, 0.09, 0.08}; Then form the array elements as the sums for(i=1; i<6; i++){ prob_interval[i]= prob_interval[i]+ prob_interval[i-1]; } Use this approach in the

Step 3 below. Important Note 2. : The array prob_interval[ ] should be generated only one time, before you begin any looping. Do not generate this array over-and-over-again, by incorrectly placing it inside the loop in Step 1. Step 3. Print out a “word” of length NL letters to the display. This “word” will be constructed from pseudorandom letters a-z. Start with a for-loop for(j=0; j a 0.085 b 0.021 c 0.045 d 0.034 e 0.112 f 0.018 g 0.025 h 0.030 i 0.075 j 0.002 k 0.011 l 0.055 m 0.030 n 0.067 o 0.07 p 0.032 q 0.002 r 0.076 s 0.057 t 0.070 u 0.036 v 0.011 w 0.013 x 0.002 y 0.018 z 0.002 ---------------- total 1.000
Use the method from Step 2. You can make use of the sequential ascii character codes for letters ‘a’ through ‘z’. float prob_let[26]= {0.085, 0.021, 0.045, 0.034, 0.112, 0.018, 0.025, 0.030, 0.075, 0.002, 0.011, 0.055, 0.030, 0.067, 0.071, 0.032, 0.002, 0.076, 0.057, 0.070, 0.036, 0.011, 0.013, 0.002, 0.018, 0.002}; Important Note: The first letter of the first word in the sentence should be printed as capitalized; i.e. uppercase. Step 4. Place a blank space after the “word” printed in Step 3, or place a period after the “word” printed in Step 3 if it is the last “word” in the sentence. Your results should look like: Elfh gfk llae mjlodp noc tjvjs mlknko si. Note, we haven’t applied any rules such as: 1. a “minimum of one vowel per word”; 2. ‘t’ is often followed by ‘h to form ‘th’; 3. ‘s’ often occurs at the end of a word to form a plural; 4. etc.. so actual words will only appear infrequently

assume that IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

Find the probability that a randomly selected person has an IQ score less than 115.

Find the probability that a randomly selected person has an IQ score greater than 118.

Find the probability that a randomly selected person has an IQ score between 88 and 112.

1. Random variable X is drawn from a normal distribution with mean 5.44 and std dev 2.54.
   1. Calculate the probability of X being less than 3.29.
   2. What is the probability of X exceeding 4.61?
   3. What is the probability of X lying between 5.79 and 7.8?
   4. Verify your answers to parts 1 2 and 3 above using numerical sampling.

Suppose that the probability that any random student graduates with honors is 0.05. Also, we know that the probability of a student that graduated with honors, getting into graduate school is 0.80. The probability that a student that did not graduate with honors gets into graduate school is 0.10. TRUE or FALSE: Given that a random student got into graduate school, it is more likely that the student graduated with honors.

True

False

A Gaussian random voltage has a mean of 4 and a variance of 10.

a.) What is the probability than an observed value of the voltage is greater than zero?

b.) What is the probability than an observed value of the voltage is greater than zero but less than or equal to the mean value?

c.) What is the probability that an observed value of the voltage is greater than twice the mean value?

About 60% of adults use cash for payments. Suppose four adults were randomly selected. a. What kind of probability is this? Explain why. b. What is the probability that one of the adults use cash for payments? c. What is the probability that more than three adults use cash for payments? d. What is the mean and variance? Interpret your results.