Multiple choice

A random variable $X$ takes the values $0,\ 1,\ 2,\ 3$ and its mean is$1.3$. If $P(X=3)=2P(X=1)$ and $P(X=2)=0.3$ , then $P(X=0)=$

  1. $0.1$
  2. $0.2$
  3. $0.3$
  4. $0.4$
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D Correct answer
Explanation

Sum of probabilities = 1. P(0) + P(1) + P(2) + P(3) = 1. P(2) = 0.3, so P(0) + P(1) + P(3) = 0.7. P(3) = 2P(1). So P(0) + 3P(1) = 0.7. Mean = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) = 1.3. P(1) + 2(0.3) + 3(2P(1)) = 1.3. P(1) + 0.6 + 6P(1) = 1.3. 7P(1) = 0.7. P(1) = 0.1. Then P(0) + 3(0.1) = 0.7, so P(0) = 0.4.

AI explanation

Let P(X=0) = a, P(X=1) = b, P(X=3) = 2b, and P(X=2) = 0.3. Because the total probability is 1, we write the equation a + b + 0.3 + 2b = 1, which simplifies to a + 3b = 0.7. The mean is calculated as (0 times a) + (1 times b) + (2 times 0.3) + (3 times 2b) = 1.3, giving 7b = 0.7. This yields b = 0.1, and substituting this into the probability sum equation gives a + 0.3 = 0.7. Therefore, P(X=0) is 0.4.

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