Using formula P(x) = x/15
For each of the values x = 1, 2, 3, 4, and 5.
a. probability distribution of x.
b. Show that the probability distribution of x satisfies the properties of a discrete probability distribution.
c. mean of x.
sketch the two specified normal curves on the same set of axes:
µ = 20 and o = 3, µ = 20 and o = 6.
a. To find the probability distribution of x using the formula P(x) = x/15, we substitute the given values of x into the formula:
For x = 1:
P(1) = 1/15 = 0.0667
For x = 2:
P(2) = 2/15 = 0.1333
For x = 3:
P(3) = 3/15 = 0.2
For x = 4:
P(4) = 4/15 = 0.2667
For x = 5:
P(5) = 5/15 = 0.3333
So, the probability distribution of x is:
x | P(x)
1 | 0.0667
2 | 0.1333
3 | 0.2
4 | 0.2667
5 | 0.3333
b. To show that the probability distribution of x satisfies the properties of a discrete probability distribution, we need to verify two conditions:
1. The sum of all probabilities is equal to 1:
0.0667 + 0.1333 + 0.2 + 0.2667 + 0.3333 = 1
2. The probabilities are non-negative:
All the probabilities in the distribution are greater than or equal to 0.
Since both conditions are satisfied, the probability distribution of x satisfies the properties of a discrete probability distribution.
c. To find the mean of x, we multiply each value of x by its corresponding probability and sum them up:
Mean (μ) = (1 * 0.0667) + (2 * 0.1333) + (3 * 0.2) + (4 * 0.2667) + (5 * 0.3333)
= 0.0667 + 0.2666 + 0.6 + 1.0668 + 1.6665
= 3.6666
So, the mean of x is 3.6666.
For the sketch of normal curves, please provide more information about what is required or the specific instructions for the sketch.
a. To find the probability distribution of x using the formula P(x) = x/15, we substitute each given value of x (1, 2, 3, 4, and 5) into the formula.
For x = 1:
P(1) = 1/15 = 0.067
For x = 2:
P(2) = 2/15 ≈ 0.133
For x = 3:
P(3) = 3/15 = 0.2
For x = 4:
P(4) = 4/15 ≈ 0.267
For x = 5:
P(5) = 5/15 = 0.333
Therefore, the probability distribution of x is:
x | P(x)
-----------------
1 | 0.067
2 | 0.133
3 | 0.2
4 | 0.267
5 | 0.333
b. To show that the probability distribution of x satisfies the properties of a discrete probability distribution, we need to check two conditions:
1. The sum of all probabilities should be equal to 1.
P(1) + P(2) + P(3) + P(4) + P(5) = 0.067 + 0.133 + 0.2 + 0.267 + 0.333 = 1
2. All probabilities should be non-negative.
All the probabilities obtained in part a are positive values, so this condition is satisfied.
Therefore, the probability distribution of x satisfies the properties of a discrete probability distribution.
c. To find the mean of x, we need to calculate the weighted average of the random variable x using the probability distribution.
Mean (µ) of x = Σ(x * P(x)), where Σ denotes the sum.
Using the values from the probability distribution:
Mean (µ) = (1 * 0.067) + (2 * 0.133) + (3 * 0.2) + (4 * 0.267) + (5 * 0.333)
= 0.067 + 0.266 + 0.6 + 1.068 + 1.665
= 3.666
Therefore, the mean of x is 3.666.
For the next part of your question, sketching the normal curves:
In this case, we have two normal curves specified by different values of µ (mean) and σ (standard deviation):
Curve 1:
µ = 20
σ = 3
Curve 2:
µ = 20
σ = 6
To sketch these curves on the same set of axes, you will need to plot the points that represent the normal distribution for each curve. The x-axis represents the x-values, while the y-axis represents the probability density.
Note that the normal distribution is symmetric around the mean (µ) and its shape is determined by the standard deviation (σ).
Start by drawing the x-axis and labeling it accordingly. Then, plot the points for each curve using the given mean and standard deviation values. The height of the curve at each x-value indicates the probability density at that point.
Make sure to label each curve appropriately so that it is clear which curve represents which set of parameters.