Question

A has four keys in his pocket, he tries to unlock a door and tries with each key until he finds the right one, let X be random variable number of keys tried including the right key to open the door,...

a) is x a discrete or continuous random variable? Explain

b) determine probability function for random variable x.

C) what is the probability that the man will try at most 2 keys before he finds the right one?

D) determine the mean and standard deviation of x.

Answer 1

a)

A random variable $X$ has possible values $1,2,3,4$ which constitute a finite set. Therefore $x$ is a discrete random variable.

b)

If unsuccessful keys are eliminated then it can take at most 4 attempts to open the door with the following probabilities:

Probability that the first key will open the door is

$$P(X=1)=1/4$$

Probability that the second key will open the door is

$$P(X=2)=(1-1/4)(1/3)=1/4$$

Probability that the third key will open the door is

$$P(X=3)=(1-1/4)(1-1/3)(1/2)=1/4$$

Probability that the fourth key will open the door is

$$P(X=4)=(1-1/4)(1-1/3)(1-1/2)(1)=1/4$$

$X$ has the uniform distribution with $p=1/4$

$$P(X=x)=1/4, x=1,2,3,4$$

c)

$$P(X\leq 2)=P(X=1)+P(X=2)$$

$$=1/4+1/4=1/2$$

d)

$$mean=E(X)=1(\dfrac{1}{4})+2(\dfrac{1}{4})+3(\dfrac{1}{4})+4(\dfrac{1}{4})=\dfrac{5}{2}$$

$$E(X^2)=1^2(\dfrac{1}{4})+2^2(\dfrac{1}{4})+3^2(\dfrac{1}{4})+4^2(\dfrac{1}{4})=\dfrac{15}{2}$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2$$

$$=\dfrac{15}{2}-(\dfrac{5}{2})^2=\dfrac{5}{4}$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{5}{4}}=\dfrac{\sqrt{5}}{2}$$