Suppose a random variable X has a uniform distribution over the interval [-10,10].
• Find the probability that X lies between -2 and 6?
• Find the probability that X2 lies between 9 and 64? (Note X2 is the square of X.)
To find the probability that a random variable X lies between -2 and 6 when it has a uniform distribution over the interval [-10,10], we need to determine the proportion of the interval that lies within the given range.
Step 1: Determine the length of the interval [-2, 6].
The length of the interval is given by 6 - (-2) = 8.
Step 2: Determine the length of the total interval [-10, 10].
The length of the total interval is given by 10 - (-10) = 20.
Step 3: Calculate the probability.
The probability is defined as the ratio of the length of the interval [-2, 6] to the length of the total interval [-10, 10].
Therefore, the probability is 8/20 = 0.4.
Hence, the probability that X lies between -2 and 6 is 0.4.
Now let's move on to finding the probability that X^2 lies between 9 and 64, where X^2 is the square of X.
Step 1: Determine the values of X that would result in X^2 lying between 9 and 64.
To do this, we need to find the square roots of 9 and 64, which are 3 and 8, respectively.
Step 2: Determine the range of X that corresponds to the square roots.
For X^2 to lie between 9 and 64, X must lie between -8 and -3, as well as between 3 and 8.
Step 3: Determine the length of each range.
The length of the first range, [-8, -3], is given by -3 - (-8) = 5.
The length of the second range, [3, 8], is given by 8 - 3 = 5.
Step 4: Determine the length of the total interval [-10, 10].
The length of the total interval is given by 10 - (-10) = 20.
Step 5: Calculate the probability.
The probability is defined as the sum of the lengths of the two ranges, [-8, -3] and [3, 8], divided by the length of the total interval [-10, 10].
Therefore, the probability is (5 + 5)/20 = 10/20 = 0.5.
Hence, the probability that X^2 lies between 9 and 64 is 0.5.