1. The probabilities that a bakery has a demand for 2, 3, 5, or 7 birthday cakes on any given day are 0.35, 0.41, 0.15, and 0.09, respectively. Create a probability distribution table for this problem Verify if it is a valid probability distribution. Draw a graph for this probability distribution. Find the mean, variance, and standard deviation.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![**Problem: Probability Distribution of Birthday Cake Demand**
1. The probabilities that a bakery has a demand for 2, 3, 5, or 7 birthday cakes on any given day are 0.35, 0.41, 0.15, and 0.09, respectively. Create a probability distribution table for this problem.
**Tasks:**
- Verify if it is a valid probability distribution.
- Draw a graph for this probability distribution.
- Find the mean, variance, and standard deviation.
**Explanation:**
- **Probability Distribution Table:**
| Number of Cakes (X) | Probability (P(X)) |
|---------------------|--------------------|
| 2 | 0.35 |
| 3 | 0.41 |
| 5 | 0.15 |
| 7 | 0.09 |
- **Validity Check:**
A probability distribution is valid if the sum of all probabilities equals 1. Verify by adding 0.35 + 0.41 + 0.15 + 0.09.
- **Graph:**
Draw a bar graph showing the number of cakes on the x-axis and their corresponding probabilities on the y-axis.
- **Calculations:**
- **Mean (Expected Value):** Calculate the mean using the formula:
\[
\mu = \sum (X \times P(X))
\]
- **Variance:** Calculate the variance using:
\[
\sigma^2 = \sum ((X - \mu)^2 \times P(X))
\]
- **Standard Deviation:** The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
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