Question

The first few problems ask you to "describe" a random variable, which means:

Give the sample space S (the result of the random experiment, from which the output of the random variable is calculated);

Give RX (you may schematize it if it is very complicated or infinite);

Give fX (you may use fractions or decimals) and show how it was calculated unless it is very simple;

Problem 2:

Suppose we have a sack with 2 red balls and 5 black balls, and we draw balls without replacement until a red ball is drawn. Let X = "the number of balls drawn".

Describe the random variable XX.

Answer 1

Scenerio:

Sack = (2R, 5B)

Balls are drawn without replacement until a R is drawn.

X = the number of balls drawn.

Trial 1: 1R is drawn. So X= 1. P(X=1) = 2/7 = 0.2857

Trial 2: 1B, 1R : So, X = 2. P(X=2) = (5/7) X (2/6) = 0.7143 X 0.3333 = 0.2381

Trial 3: 1B, 1B, 1R. S, X = 3. P(X=3) = (5/7) X (4/6) X (2/5) =0.7143 X 0.0.6667 X 0.4 = 0.1905

Trial 4: 1B, 1B, 1B, 1R. So, X = 4. P(X=4) = (5/7) X (4/6) X (3/5) X (2/4) = 0.7143 X 0.6667 X 0.6 X 0.5 = 0.1429

Trial 5: 1B, 1B, 1B, 1B, 1R, X = 5. P(X =5) = (5/7) X (4/6) X (3/5) X (2/4) X (2/3) = 0.7143 X 0.6667 X 0.6 X 0.5 X 0.6667 = 0.0953

Trial 6: 1B, 1B, 1B, 1B, 1B, 1R:X = 6. P(X=6) = (5/7) X (4/6) X (3/5) X (2/4) X (1/3) X (2/2) = 0.7143 X 0.6667 X 0.6 X 0.5 X 0.3333 = 0.0476

Thus, we note:

X can take values: 1,2,3,4,5 and 6 with the following probabilities:

X              P

1            0.2857

2           0.2381

3             0.1905

4             0.1429

5            0.0953

6           0.0476

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Total        1