Two more PMF/joint PMF questions

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  • Thread starter nacho-man
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In summary, the author provides an expansion for $\displaystyle e^{-\lambda}$ and $\displaystyle e^{-\lambda}$ which gives the result that $e^{-\lambda}=1$. They also provide an equation for $P\{Y=k\}$ which depends on $a$, $k$, and $0$.
  • #1
nacho-man
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Please refer to the attached images.

For question 2,
How does this converge to one? i tried using a ratio test but thought it converged to zero

For question 3,
I understand how they get a $\frac{1}{1+b-a}$ in the denominator, but not how they set up the rest.
for example if Y = min(0,X)
then Y>0 is impossible because the min(0,X) restricts the value of Y to 0. Thus making $P(Y>0) = 0$
Similarly, $P(Y<a) = 0$ since the bounds are restricted from $a<0<b$.

for $y=0$ this can occur when X takes any value over $[0,b]$. There are $b+1$ such values.

However, for $a<y<0$, aren't there $0-a+1$ such values? Why do they simply have a $1$ in the numerator.
 

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  • #2
nacho said:
Please refer to the attached images.

For question 2,
How does this converge to one? i tried using a ratio test but thought it converged to zero

Remembering the well known expansion...

$\displaystyle e^{\lambda} = \sum_{n=0}^{\infty} \frac{\lambda^{n}}{n!}\ (1)$

... it is immediate to write...

$\displaystyle e^{- \lambda}\ \sum_{n=0}^{\infty} \frac{\lambda^{n}}{n!}= e^{-\lambda}\ e^{\lambda} = 1\ (2)$

Kind regards

$\chi$ $\sigma$
 
  • #3
nacho said:
For question 3,
I understand how they get a $\frac{1}{1+b-a}$ in the denominator, but not how they set up the rest.
for example if Y = min(0,X)
then Y>0 is impossible because the min(0,X) restricts the value of Y to 0. Thus making $P(Y>0) = 0$
Similarly, $P(Y<a) = 0$ since the bounds are restricted from $a<0<b$.

for $y=0$ this can occur when X takes any value over $[0,b]$. There are $b+1$ such values.

However, for $a<y<0$, aren't there $0-a+1$ such values? Why do they simply have a $1$ in the numerator.

If X is uniformely distributed in $a \le X \le b$ and is $a < 0$ then... $\displaystyle P \{X = k \} = \frac{1}{1 + b - a}\ \forall k\ \text{with}\ a \le k \le b\ (1)$

If $Y = \text{min}\ (0,X)$ then is $\displaystyle P \{Y = k \} = \frac{1}{1 + b - a}\ \forall k\ \text{with}\ a \le k \le - 1$, $\displaystyle P \{Y = k \} = \frac{1+b}{1 + b - a}\ \text{if}\ k =0$ and 0 for all other k... Kind regards $\chi$ $\sigma$
 
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FAQ: Two more PMF/joint PMF questions

What is a PMF?

A PMF, or Probability Mass Function, is a function that describes the probability of a discrete random variable taking on a certain value. It shows the distribution of possible outcomes and their associated probabilities.

What is a joint PMF?

A joint PMF, or Joint Probability Mass Function, is a function that describes the probability of two or more discrete random variables taking on specific values simultaneously. It shows the distribution of all possible combinations of outcomes and their associated probabilities.

How do you calculate a PMF?

To calculate a PMF, you need to first identify all possible outcomes of the discrete random variable. Then, assign probabilities to each outcome based on the given information. Finally, add up all the probabilities to ensure they sum to 1, as the total probability of all possible outcomes must be 1.

How is a joint PMF different from a marginal PMF?

A joint PMF shows the probabilities of multiple discrete random variables occurring simultaneously, while a marginal PMF shows the probabilities of a single discrete random variable occurring on its own, ignoring the other variables. In other words, a joint PMF considers all possible combinations of outcomes, while a marginal PMF only considers one variable at a time.

What is the relationship between a joint PMF and a conditional PMF?

A conditional PMF is calculated by dividing the joint PMF by the marginal PMF of the given condition. It shows the probability of a specific outcome of one variable, given a specific outcome of another variable. In other words, it shows how the probability of one variable is affected by the outcome of another variable.

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