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.)
Well, well, well, if it isn't my old friend, probability! Let's crunch some numbers, shall we?
For the first part, we have a uniform distribution from -10 to 10. So the total width of this interval is 20.
Now we want to find the probability that X lies between -2 and 6. Well, that's a width of 6 - (-2) = 8.
Since we have a uniform distribution, the probability is simply the width of our desired interval divided by the total width of the interval. So, the probability is 8/20 = 2/5.
Now, for the second part, we need to find the probability that X^2 lies between 9 and 64. To get the probability, we can think of it as finding the probability that X lies between -√64 and √64. As we know, -√64 is -8 and √64 is 8.
So, we want X to lie between -8 and 8. The width of this interval is 8 - (-8) = 16.
Again, using the same logic, the probability is 16/20 = 4/5.
I hope I haven't squared your hopes and dreams with my square answers!
To find the probability that a uniformly distributed random variable X lies between -2 and 6, we can use the formula for the uniform probability density function:
1. The probability density function (pdf) for a uniform distribution is given by:
f(x) = 1 / (b - a) for a <= x <= b
where a and b are the lower and upper limits of the interval.
In this case, a = -10 and b = 10, so the pdf is:
f(x) = 1 / (10 - (-10)) = 1 / 20
2. To find the probability that X lies between -2 and 6, we need to calculate the integral of the pdf function over this interval:
P(-2 <= X <= 6) = ∫[from -2 to 6] f(x) dx
= ∫[from -2 to 6] (1 / 20) dx
3. Simplifying the integral, we get:
P(-2 <= X <= 6) = (1 / 20) * (x) [from -2 to 6]
= (1 / 20) * (6 - (-2))
= (1 / 20) * 8
= 8 / 20
= 0.4
Therefore, the probability that X lies between -2 and 6 is 0.4.
Now, let's move on to finding the probability that X^2 (the square of X) lies between 9 and 64.
4. Since X is uniformly distributed in the interval [-10, 10], its square, X^2, is also non-negative and distributed within the interval [0, 100].
5. We need to find the probability that X^2 lies between 9 and 64. To do this, we subtract the probability that X^2 is less than 9 from the probability that X^2 is less than or equal to 64:
P(9 <= X^2 <= 64) = P(X^2 <= 64) - P(X^2 < 9)
6. To find the probability that X^2 is less than or equal to 64, we calculate the cumulative distribution function (cdf) for X^2 at 64:
P(X^2 <= 64) = P(X <= 8) (since the square root of 64 is 8)
7. Using the uniform pdf formula, we can find the probability of X being less than or equal to 8:
P(X <= 8) = ∫[from -10 to 8] f(x) dx
= ∫[from -10 to 8] (1 / 20) dx
= (1 / 20) * (x) [from -10 to 8]
= (1 / 20) * (8 - (-10))
= (1 / 20) * 18
= 18 / 20
= 0.9
8. Similarly, to find the probability that X^2 is less than 9, we calculate the cdf for X^2 at 9:
P(X^2 < 9) = P(X < 3) (since the square root of 9 is 3)
9. Using the uniform pdf formula, we calculate the probability of X being less than 3:
P(X < 3) = ∫[from -10 to 3] f(x) dx
= ∫[from -10 to 3] (1 / 20) dx
= (1 / 20) * (x) [from -10 to 3]
= (1 / 20) * (3 - (-10))
= (1 / 20) * 13
= 13 / 20
10. Finally, we can find the probability that X^2 lies between 9 and 64 by subtracting the probability of X^2 being less than 9 from the probability of X^2 being less than or equal to 64:
P(9 <= X^2 <= 64) = P(X^2 <= 64) - P(X^2 < 9)
= 0.9 - (13 / 20)
= 18 / 20 - 13 / 20
= 5 / 20
= 0.25
Therefore, the probability that X^2 lies between 9 and 64 is 0.25.
To find the probability that a random variable X lies between -2 and 6, we need to calculate the area under the probability density function (PDF) between these values.
1. Understand the uniform distribution:
A uniform distribution is a continuous probability distribution where the probability of any point in the interval is equal. For a uniform distribution over the interval [-10, 10], the PDF is a rectangle with a width of 10 and a height of 1/20 (since the total area should be 1).
2. Calculate the probability:
To find the probability that X lies between -2 and 6, we need to find the area under the PDF between these values. The width of the interval is 6 - (-2) = 8, and the height of the PDF is 1/20. Therefore, the probability can be found by calculating the area of a rectangle:
Probability = width * height = 8 * (1/20) = 8/20 = 0.4
So, the probability that X lies between -2 and 6 is 0.4.
To find the probability that X^2 lies between 9 and 64, we need to consider the transformation of the random variable X^2.
3. Calculate the probability:
Since X^2 can take on any value between 0 and 100 (square of the range [-10, 10]), the probability of X^2 lying between 9 and 64 can be calculated by finding the area under the PDF between the square roots of these values.
The square root of 9 is 3, and the square root of 64 is 8. Therefore, we need to find the probability that X lies between -3 and -8 (negative values) and X lies between 3 and 8 (positive values).
To find the probability for each interval, we use the same formula as before, since the PDF remains the same:
Probability = width * height
For X in [-3, -8]:
Width = -8 - (-3) = -5
Height = 1/20
Probability (X in [-3, -8]) = (-5) * (1/20) = -5/20 = -0.25
Since probabilities cannot be negative, the probability for this interval is 0.
For X in [3, 8]:
Width = 8 - 3 = 5
Height = 1/20
Probability (X in [3, 8]) = 5 * (1/20) = 5/20 = 0.25
The total probability that X^2 lies between 9 and 64 is the sum of the probabilities for each interval: 0 + 0.25 = 0.25