Question

⦁ Mr. A likes playing a game and the probability that he wins this game is p. He enters the casino and he promises himself that he plays the game until he wins one time and then he stops. Let X be the number of plays in order to win one time. ⦁ What are the values of X? ⦁ What is the probability that X=n?. Prove that it satisfies the PMF conditions. ⦁ Calculate E(X) ⦁ Calculate V(X) ⦁ Study the memoryless property of X.

Answer 1

The possible values of X for this game are 0, 1, 2, 3, 4.......n, where n ≥ 1

How to determine the values of X?

From the complete question, we understand that Mr. A wants to plays the game until he wins

This means that

He might win at the first game and he might win after n attempts

So, the values of X are

X = 0, 1, 2, 3, 4.......n

Hence, the possible values of X for this game are 0, 1, 2, 3, 4.......n, where n ≥ 1

The probability that X = n

The probability of x is represented as:

P(x) = nCx * p^x * (1 - p)^(n-x)

So, the probability that X = n is:

P(n) = nCn * p^n * (1 - p)^(n - n)

Evaluate the exponent

P(n) = nCn * p^n * 1

Evaluate the combination expression

P(n) = 1 * p^n * 1

This gives

P(n) = p^n

Hence, the probability that X = n is p^n

Prove that it satisfies the PMF conditions.

The distribution satisfies PMF conditions because

• The sum of the probabilities is 1
• No probability is negative
• Each probability value is between 0 and 1 (inclusive)

Calculate E(X)

The expected value E(x) is calculated using

E(x) = n * p

So, we have:

E(x) = np

Hence, the value of E(x) is np

Calculate V(X)

The variance V(x) is calculated using

V(x) = √n * p * (1 - p)

So, we have:

V(x) = √np(1 - p)

Hence, the value of V(x) is √np(1 - p)

Study the memoryless property of X.

The memoryless property of X is that each probability of X is independent

Read more about probability at:

https://brainly.com/question/25870256

#SPJ1