Question

Suppose we have a sack with 2 red balls and 2 black balls, and we draw balls without replacement until the second red ball is drawn.

Describe the random variable ? = "the number of balls drawn".

Describe by giving the range, probability distribution, expected value, standard deviation, and variance.

Answer 1

Let R be an event that a red ball is drawn, B be an event that a black ball is drawn. Therefore the outcomes here are given as: (drawing till second red ball is drawn)
RR, BRR, RBR, BBRR, BRBR, RBBR

Therefore from the above sample space, the random variable X would have the following PDF:

P(X = 2) = 1/6
P(X = 3) = 2/6 = 1/3
P(X = 4) = 3/6 = 0.5

Therefore the range of X is computed here as:
Range = Max - Min values = 4 - 2 = 2
Therefore 2 is the required range here.

The PDF is already computed above for X as:
P(X = 2) = 1/6
P(X = 3) = 2/6 = 1/3
P(X = 4) = 3/6 = 0.5

The expected value of X is computed here as:

Therefore expected value of X is 3.3333

The second moment is first computed here as:

Therefore the variance here is computed as:
Var(X) = E(X²) - [E(X)]² = 11.6667 - 3.3333² = 5/9
Therefore Variance of X here is 5/9 = 0.5556

The standard deviation of X is computed here as:

Therefore 0.7454 is the standard deviation of X here.