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MATH 337: Assignment 1

2021

1. (a) Show that e α ≥ 1 + x for all real values of x.

(b) Assuming that a year has 365 days, show that in a group of 23 people, the probability that at least two people have the same birthday is greater than 1/2. [Hint: ﬁrst calculate the probabil- ity that in a group of n people, no two share the same birthday using (a).]

2. Prove that

〇 = 一 tanh ³ … ╱ ← ,

where ∆ = b 〇一 4ac > 0.

3. Prove that for |x| < 1,

兰 k 〇 x 亿 ³ … = 1 + x

亿 2←

4. Let X be a random variable with geometric distribution having pa- rameter p. That is, P (X = k) = p(1 一 p) 亿 ³ … . Show that

E(X 〇 ) =

2 一 p

p 〇 .

5. Let X be a random variable with the uniform distribution on the interval [a, b]. That is, X has density function fx given by

fx (x) = ,0(1)/(b 一

Show that X has mean (a + b)/2 and variance (b 一 a) 〇 /12.

6. If X is a random variable with density function fx (y) = e ³ g/9 , y > 0,

then X has variance θ 〇 .

7. If X and Y are random variables with the Poisson distribution of parameters λ and τ respectively, show that X + Y is again Poisson with parameer λ + τ .

8. If X is a random variable with density function fx (y) = ya ³ … e ³ g/8 , y > 0,

show that Var(X) is αβ 〇 .

9. Let Γ (x) denote the Γ -function. Show that Γ (1/2) = ′ π .

10. Let X be a random variable with the normal distribution e ³ α 2 /〇 .

Show that Var(X)=1.