Question 1 [20 marks]. This question demonstrates a special feature of the

geometric distribution that makes it suitable for applications to discrete waiting times.

It is called the memoryless property. This is a characteristic of the geometric

random variable that might or might not correspond to what happens in real-life

applications.

Assume that X models the first day that a server will fail. On each day the server fails

with probability p, independently of the other days. Equivalently, X ∼ Geom(p).

(a) For a positive integer, n1, using the known expression for the p.m.f. of X, prove

that

P(X > n1) = (1 − p)

n1

.

Deduce an expression for the c.d.f. P(X ≤ n1).

Hint: Use the formula for the sum of a geometric series 1 + r + r

2

· · · for suitable

r.

geometric distribution that makes it suitable for applications to discrete waiting times.

It is called the memoryless property. This is a characteristic of the geometric

random variable that might or might not correspond to what happens in real-life

applications.

Assume that X models the first day that a server will fail. On each day the server fails

with probability p, independently of the other days. Equivalently, X ∼ Geom(p).

(a) For a positive integer, n1, using the known expression for the p.m.f. of X, prove

that

P(X > n1) = (1 − p)

n1

.

Deduce an expression for the c.d.f. P(X ≤ n1).

Hint: Use the formula for the sum of a geometric series 1 + r + r

2

· · · for suitable

r.

Original post by theoneandonlyyo

Question 1 [20 marks]. This question demonstrates a special feature of the

geometric distribution that makes it suitable for applications to discrete waiting times.

It is called the memoryless property. This is a characteristic of the geometric

random variable that might or might not correspond to what happens in real-life

applications.

Assume that X models the first day that a server will fail. On each day the server fails

with probability p, independently of the other days. Equivalently, X ∼ Geom(p).

(a) For a positive integer, n1, using the known expression for the p.m.f. of X, prove

that

P(X > n1) = (1 − p)

n1

.

Deduce an expression for the c.d.f. P(X ≤ n1).

Hint: Use the formula for the sum of a geometric series 1 + r + r

2

· · · for suitable

r.

geometric distribution that makes it suitable for applications to discrete waiting times.

It is called the memoryless property. This is a characteristic of the geometric

random variable that might or might not correspond to what happens in real-life

applications.

Assume that X models the first day that a server will fail. On each day the server fails

with probability p, independently of the other days. Equivalently, X ∼ Geom(p).

(a) For a positive integer, n1, using the known expression for the p.m.f. of X, prove

that

P(X > n1) = (1 − p)

n1

.

Deduce an expression for the c.d.f. P(X ≤ n1).

Hint: Use the formula for the sum of a geometric series 1 + r + r

2

· · · for suitable

r.

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