Question

1.verify that f(x)=2x/k(k+1) for x=0,1,2,.....k. can serve as a pmf of a random variable x

2.A fair coin is flipped until a head appears. Let N represent the number of tosses required to realise a head. Find the pmf of N

3.The pmf of a discrete random variable X is given by p(X=x)=kx for x=1,2,3,4,5,6.

Find the value of the constant k, p(X<4) and p(3<x<6)

4.A die is loaded such the probability that of a face showing up is proportional to the face number. Determine the probability of each sample point

5.Roll a fair die and let X be the square of the show up.Write the probability distribution of x hence compute p(X<15) and p(3<x<30)

6.Let x be a random variable the number of hours observed when two dice are rolled together once. show that x is a discrete random variable.

7.For each of the following determine c so that the function can serve as a pmf of a random variable X

1. F(X)=cx for x=1,2,3,4,5
2. f(x)=cx² for x=0,1,2,...k
3. f(x)=c(1/6)^x for x=0,1,2,3,...
4. f(x)=c2^-x for x=0,1,2,....

Answer 1

1.

$$f(x)=x(\dfrac{2}{k(k+1)})\geq 0, x=0,1,2,...$$

$$\displaystyle\sum_{x=0}^kf(x)=\displaystyle\sum_{x=0}^kx(\dfrac{2}{k(k+1)})$$

$$=\dfrac{2}{k(k+1)}(\dfrac{k(k+1)}{2})=1$$

$f(x)=x(\dfrac{2}{k(k+1)})\geq 0, x=0,1,2,...$ can serve as a pmf of a random variable $x.$

2.

A geometric random variable X with parameter p has probability mass function

$$f(N) = p(1− p) ^N, N = 0,1,2,... .$$

Given $p=0.5$

$$f(N) = 0.5(1− 0.5) ^N, N = 0,1,2,... .$$

$$f(N) = 0.5 ^{N+1}, N = 0,1,2,... .$$

3.

$$k(1)+k(2)+k(3)+k(4)+k(5)+k(6)=1$$

$$k=\dfrac{1}{21}$$

$$P(X<4)=\dfrac{1}{21}(1+2+3)=\dfrac{2}{7}$$

$$P(3<X<6)=\dfrac{1}{21}(4+5)=\dfrac{3}{7}$$

4.

$$k+2k+3k+4k+5k+6k=1$$

$$k=\dfrac{1}{21}$$

$$P(X=1)=\dfrac{1}{21}$$

$$P(X=2)=\dfrac{2}{21}$$

$$P(X=3)=\dfrac{1}{7}$$

$$P(X=4)=\dfrac{4}{21}$$

$$P(X=5)=\dfrac{5}{21}$$

$$P(X=6)=\dfrac{2}{7}$$

5.

$$\begin{matrix} x & 1 & 4 & 9 & 16 & 25 & 36 \\ p(x) & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6 \end{matrix}$$

$$p(X<15)=\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{6}=\dfrac{1}{2}$$

$$p(3<X<30)=\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{6}=\dfrac{2}{3}$$

6.A discrete random variable is an rv whose possible values either constitute a

finite set or else can be listed in an infinite sequence in which there is a first

element, a second element, and so on (“countably” infinite).

A random variable the number of hours observed when two dice are rolled together once is

 countable number of possible values.

Therefore this random variable is discrete.

7.

1.

$$c+2c+3c+4c+5c=1$$

$$c=\dfrac{1}{15}$$

2.

$$\displaystyle\sum_{x=0}^{k}cx^2=\dfrac{ck(k+1)(2k+1)}{6}=1$$

$$c=\dfrac{1}{k(k+1)(2k+1)}$$

3.

$$\displaystyle\sum_{x=0}^{\infin}c(\dfrac{1}{6})^x=\dfrac{c}{1-\dfrac{1}{6}}=\dfrac{6c}{5}=1$$

$$c=\dfrac{5}{6}$$

4.

$$\displaystyle\sum_{x=0}^{\infin}c(2)^{-x}=\displaystyle\sum_{x=0}^{\infin}c(\dfrac{1}{2})^x=\dfrac{c}{1-\dfrac{1}{2}}=2c=1$$

$$c=\dfrac{1}{2}$$