For each probability and percentile problem, draw the picture.
A random number generator picks a number from 1 to 9 in a uniform manner.
1) Give the distribution of X.
X =
2) Enter an exact number as an integer, fraction, or decimal.
f(x) = where <=x=>
3) Enter an exact number as an integer, fraction, or decimal.
μ =
4)σ =
5)P(3.25 < X < 7.5) =
6)P(X > 3.67) =
7)P(X > 6 | X > 3) =
8)Find the 80th percentile. (Round your answer to one decimal place.)
Thank you
To answer the probability and percentile problems, we will first need to understand the distribution of X. Since the random number generator picks a number from 1 to 9 in a uniform manner, the distribution of X will be a discrete uniform distribution.
1) The distribution of X:
X = {1, 2, 3, 4, 5, 6, 7, 8, 9}
To visualize this distribution, you can draw a simple bar graph with the values of X on the x-axis and the probabilities on the y-axis. Each bar should have equal height (1/9) since the distribution is uniform.
2) To find the probability mass function (PMF) of X, we need to assign probabilities to each value of X. Since the distribution is uniform, each value has an equal probability of occurring.
f(x) = 1/9, where 1 <= x <= 9
3) The mean (μ) of a uniform distribution can be found by taking the average of the minimum and maximum values.
μ = (1 + 9) / 2 = 5
4) The standard deviation (σ) of a discrete uniform distribution can be found using the following formula:
σ = √((n^2 - 1) / 12), where n is the number of possible outcomes (in this case, n = 9).
σ = √((9^2 - 1) / 12) = √(80/12) ≈ 2.581
Now let's move on to the probability and percentile questions.
5) To find P(3.25 < X < 7.5), we need to calculate the cumulative probability for each value of X within that range and subtract the lower cumulative probability from the higher cumulative probability.
P(3.25 < X < 7.5) = P(X < 7.5) - P(X < 3.25)
Since the distribution is uniform, we can find the cumulative probability simply by dividing the number of values within the given range by the total number of values.
P(3.25 < X < 7.5) = (7 - 3) / 9 = 4/9 ≈ 0.4444
6) To find P(X > 3.67), we need to calculate the cumulative probability for X being less than or equal to 3.67 and subtract it from 1.
P(X > 3.67) = 1 - P(X <= 3.67)
Since the distribution is uniform, we can find the cumulative probability for X being less than or equal to 3.67 by dividing the number of values less than or equal to 3.67 by the total number of values.
P(X > 3.67) = 1 - (3 / 9) = 6/9 ≈ 0.6667
7) To find P(X > 6 | X > 3), we need to calculate the conditional probability of X being greater than 6 given that it is already greater than 3.
P(X > 6 | X > 3) = P(X > 6 and X > 3) / P(X > 3)
Since the distribution is uniform, the probability of X being greater than 6 and X being greater than 3 is the same as the probability of X being greater than 6.
P(X > 6 | X > 3) = P(X > 6) / P(X > 3)
Using the same reasoning as before, we can calculate the probabilities:
P(X > 6 | X > 3) = (9 - 6) / (9 - 3) = 3/6 = 0.5
8) To find the 80th percentile, we need to find the value of X such that 80% of the distribution lies below that value.
Since the distribution is uniform, we can calculate the percentile by multiplying the total number of values by the desired percentile (0.80 in this case) and rounding it up to the nearest whole number.
80th percentile = 9 * 0.80 = 7.2 (rounded up to 8)
Therefore, the 80th percentile is 8.
I hope this explanation helps you understand how to solve probability and percentile problems using the given distribution of X. If you have any further questions, feel free to ask!