1.Let 𝑋 and 𝑌 be two binomial random variables:

Question

a.If 𝑋 and 𝑌 are independent, then 𝑋+𝑌 is also a binomial random variable
b.If 𝑋 and 𝑌 have the same parameters, 𝑛 and 𝑝 , then 𝑋+𝑌 is a binomial random variable
c.If 𝑋 and 𝑌 have the same parameter 𝑝 , and are independent, then 𝑋+𝑌 is a binomial random variable.
2.Suppose that, 𝐄[𝑋]=0 . Then, 𝑋=0 .
3.Suppose that, 𝐄[𝑋^2]=0 . Then, 𝐏(𝑋=0)=1 .
4.Let 𝑋 be a random variable. Then, 𝐄[𝑋^2]≥𝐄[𝑋]. True or false?
5.Suppose that, 𝑋 is a random variable, taking positive integer values, which satisfies 𝐄[(𝑋−6)^2]=0 . Then, 𝑝𝑋(4)=𝑝𝑋(5) .
6.Suppose that 𝐄[𝑋]≥0 . Then, 𝑋≥0 with probability 1, i.e., 𝐏(𝑋≥0)=1.
True or false?

Answer 1

1.
a) False
b) False
c) True
2. False
3. True

Answer 2

1.a False
1.b False
1.c True
2. False
3. True
4. False
5. True
6. False