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Chapter 3: Problem 16

Let the mutually independent random variables \(X_{1}, X_{2}\), and \(X_{3}\) be \(N(0,1)\), \(N(2,4)\), and \(N(-1,1)\), respectively. Compute the probability that exactly two of these three variables are less than zero.

Short Answer

Expert verified
The probability that exactly two of these three variables are less than zero is the sum of the probabilities of the individual combinations obtained in Step 3, which results in approximately 0.2116.

Step by step solution

01

Calculate individual probabilities

This step involves calculating the probabilities of each individual variables being less than zero. This is done using the cumulative distribution function of the normal distribution given by the expression \( \Phi( \frac{x - 渭}{蟽}) \) which gives the probability that a normally distributed random variable \(X\) is less than or equal to \(x\), where \(渭\) and \(蟽\) are the mean and standard deviation of the normal distribution respectively. For \(X_1\), \(X_2\), \(X_3\), use respective \(渭\) and \(蟽\) values to get \(P[X_1 < 0] = \Phi(-\frac{0}{1})\), \(P[X_2 < 0] = \Phi(-\frac{2}{2})\), and \(P[X_3 < 0] = \Phi(\frac{1}{1})\).
02

Use standard normal tables to find the probabilities

Look up the above calculated values in a standard normal (Z) table to get the probabilities. Using the table you would find \(P[X_1 < 0] = 0.5\), \(P[X_2 < 0] = 0.1587\), and \(P[X_3 < 0] = 0.8413\)
03

Calculate respective probabilities for 2 out of 3 variables being less than zero

There are three ways in which exactly 2 out of 3 variables can be less than zero. Calculate the probability for each combination. Use the formula \(P(A \cap B \cap \stackrel{-}{C} ) = P(A)P(B)(1 - P(C))\) where \(A \cap B \cap \stackrel{-}{C}\) refers to the event that A and B occur and C does not occur. The individual combinations are as follows: \(P[X_1 < 0, X_2 < 0, X_3 鈮� 0] = (0.5)(0.1587)(1 - 0.8413)\) \(P[X_1 < 0, X_2 鈮� 0, X_3 < 0] = (0.5)(1 - 0.1587)(0.8413)\) \(P[X_1 鈮� 0, X_2 < 0, X_3 < 0] = (1 - 0.5)(0.1587)(0.8413)\)
04

Sum up the probabilities

Sum the probabilities from the combinations calculated in step 3 to get the final probability of exactly 2 out of the three variables being less than zero.

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