1. Let X follow a geometric distribution. Show that P(X = k) = 1.
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!
![### Educational Content: Probability and Statistics Problems
#### Problem 1:
Let \( X \) follow a geometric distribution. Show that
\[
\sum_{k=1}^{\infty} P(X = k) = 1.
\]
#### Problem 2:
A class consists of 18 boys and 16 girls. The teacher will randomly choose three students to be on a special committee. What is the probability that more girls are on the committee than boys?
#### Problem 3:
Jameson is really bored. He is rolling a pair of fair six-sided dice. He will keep rolling the dice until he has rolled a seven (i.e., the sum of the dice is seven) three times. What is the probability that he will need to roll the dice 8, 9, or 10 times?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F060003b4-6213-43cc-bee0-06286043c8c7%2Fa8aa2d79-e9c4-4595-8810-2a4e47e65935%2Fr1or1j2_processed.jpeg&w=3840&q=75)
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