TUTORIAL QUESTION 1
STQS2013
1. Let 𝑋 be the outcome when a fair 6-sided die is rolled.
a) Find the probability distribution function of 𝑋.
b) Find the cumulative distribution function of 𝑋.
c) Evaluate the following:
i. 𝑃(𝑋 ≤ 3)
ii. 𝐹(5)
iii. 𝑃(2 < 𝑋 ≤ 6)
2. Let 𝑋 be the number of heads obtained when two fair coins are tossed.
a) Find the probability distribution function of 𝑋.
b) Find the cumulative distribution function of 𝑋.
c) Evaluate the following:
i. 𝑃(𝑋 ≤ 1)
ii. 𝐹(2)
iii. 𝑃(𝑋 = 2)
3. Let 𝑋 be a discrete random variable with the following probability mass function:
𝑓(𝑥) = {
𝑘(𝑥 + 1), = 0,1,2,3
0, otherwise
a) Find the value of 𝑘 that makes this a valid probability distribution function.
b) Write the complete pdf with the correct value of 𝑘.
c) Find the cumulative distribution function of 𝑋.
4. Let 𝑋 be a discrete random variable with the following distribution:
x
2
4
6
8
10
𝑓(𝑥)
k
2k
3k
2k
k
Find the value of 𝑘 that makes this a valid probability distribution function.
a) Find the value of 𝑘 that makes this a valid probability distribution function.
b) What is 𝑃(𝑋 = 6)?
c) What is 𝑃(𝑋 ≥ 6)?
d) Find the distribution function
5. Let 𝑋 be a discrete random variable with the following cumulative distribution function:
𝐹(𝑥) =
a) Find the probability distribution function of 𝑋.
b) Evaluate the following:
i. 𝑃(𝑋 = 4)
ii. 𝑃(𝑋 > 6)
iii. 𝑃(𝑋 ≤ 8)
TUTORIAL QUESTION 2
STQS2013
1. Let 𝑋 be a discrete random variable with the following cumulative distribution function:
𝐹(𝑥) =
a) Find the quantile function of 𝑋.
b) Compute the first quartile, median and third quartile.
2. Let 𝑋 be a discrete random variable with the following probability distribution function:
𝑓(𝑥) =
a) Find the cumulative distribution function of 𝑋.
b) Find the quantile function of 𝑋.
c) Compute the first deciles, 𝑝 = 0.35 and 𝑝 = 0.8.
3. Let 𝑋 be a continuous random variable with the probability density function given by:
𝑓(𝑥) =
a) Find the cumulative distribution function of 𝑋.
b) Evaluate the following:
i. 𝑃(0.2 < 𝑋 < 0.6)
ii. 𝑃(𝑋 > 0.5)
c) Find the quantile function of 𝑋. Then, compute the median and the 90th percentile.
4. Let 𝑋 be a continuous random variable with the probability density function given by:
𝑓(𝑥) =
a) Find the value of k.
b) Find the cumulative distribution function of 𝑋.
c) Evaluate the following:
i. 𝑃(𝑋 ≤ 0.5)
ii. 𝑃(−0.5 < 𝑋 ≤ 0.5)
5. Let 𝑋 be a continuous random variable with the following cumulative distribution function:
𝐹(𝑥) =
a) Find the quantile function of 𝑋.
b) Evaluate the following:
i. 𝑃(𝑋 ≤ 1)
ii. 𝑃(𝑋 > 2)
ii. 𝑃(1 < 𝑋 ≤ 2)
c) Determine the probability density function of 𝑋.