Cumulative Probabilities Worksheet

There are some examples you may wish to examine before attempting these exercises.

Note

Interpret <= as less than or equal to (at most, or no more than).

Exercise 1

Here's the probability distribution for a discrete random variable X

x f(x) P[ X <= x ]
0 0.15  
1 0.35  
2 0.25  
3 0.15  
4 0.10  
  1. Find the mean and standard deviation for this variable.
  2. Compute and table the cumulative probabilities in the third column of the table.

Exercise 2

Here's the probability distribution for X = number of children in a randomly selected family. The mean is 1.6094. Compute the and table the cumulative probabilities in the third column.

x f(x) P[ X <= x ]
0 0.20000  
1 0.32189  
2 0.25903  
3 0.13896  
4 0.05591  
5 0.01800  
6 0.00483  
7 0.00111  
8 0.00022  
9 0.00004  
10 0.00001  

Exercise 3

Based on recent records, the manager of a car painting center has determined the following cumulative probabilities for X = the number of customers per day.

x P[ X <= x ]
0 0.05
1 0.25
2 0.55
3 0.80
4 0.95
5 1.00
  1. What is the probability the center serves at most 3 customers in a day?
  2. What is the probability the center serves less than 3 customers in a day?
  3. What is the probability the center serves exactly 3 customers in a day?
  4. What is the probability the center serves at most 3 customers in a day?
  5. What is the probability the center serves more than 3 customers in a day?
  6. What is the probability the center serves no customers in a day?
  7. Find the probability X is between 2 and 4 inclusive. That is, find P[2<=X<=4].
  8. Find P[ 1<=X<=3 ].

Exercise 4

When a fair coin is tossed 6 times, and X = number of heads is counted, then the cumulative probabilities are as tabled below.

x P[ X <= x ] f(x)
0 0.0156  
1 0.1094  
2 0.3438  
3 0.6563  
4 0.8906  
5 0.9844  
6 1.0000  
  1. Find the probability there are at most 4 heads.
  2. Find the probability there are exactly 4 heads.
  3. Find the probability there are less than 4 heads.
  4. Find the probability there are more than 4 heads.
  5. Find the probability there are at least 4 heads.
  6. Find the probability there are between 1 and 5 heads, inclusive.
  7. What's the probability each of the 6 tosses comes up the same?
  8. Look on page 657 of your textbook. Use the row labeled n = 6 (there are n = 6 trials, or tosses of the coin) and the column labeled p = 0.50 (the probability of a head in any one trial is p = 0.50). Compare to the table above.
  9. Fill in the column of probabilities f(x). Sum these.
  10. Find the mean of X. Verify that this makes sense. Rumor has it that there's a shortcut for computing the mean in this circumstance-merely find np. Is this true?
  11. Find the variance and standard deviation of X. Rumor has it that there's a shortcut for computing the variance in this circumstance-merely find np(1 - p). Is this true?