Joint probability mass problem

Question

Additional information: Let $q_x$ denote the probablity of a $x$ year old person who's going to die within 1 year. Explain cumulative distribution function of $X$ is given by:

$$F(x)=P(X\leq x)=1-\prod^{x}_{y=0}(1-q_{35+y})$$

letting $q_x$ denote the probablity of that a $x$ year old person is going to die before the person is able to become $x+1$ year old. That means that the chance of the person surviving this year is $1-q_x$. Notice that vi can write

$$F(x)=P(X\leq x) = 1-P(X>x)$$

where $P(X>x)$ is the probablity for that the person is becoming older than $x$. This means that the person has survived up till $x$ years. We're interested in when they are over 35 years.

$$\prod^x_{y=0}(1-q_{35+y})$$

$$\therefore{F(x)=1-P(X>x)=1-\prod^x_{y=0}(1-q_{35+y})}$$

Question: How do I show that the join probabilty mass function is given by

$$F(x)=P(X\leq x)=1-P(X>x)$$

for $x=0,1,2,3,...71$

is this any correct?

$$F(x)-F(x-1)=P(X\leq x) - P(X\leq x-1)$$

$$=1-\prod^x_{y=0}(1-q_{35+y})-1-\prod^{x-1}_{y=0}(1-q_{35+y})$$

$$-\bigg(\prod^{x}_{y_0}(1-q_{35+y})+\prod^{x-1}_{y_0}(1-q_{35+y})\bigg) $$

and im stuck here, what do i do next?

Answer 1

We are not told what is $X$, but we are given a formula for its CDF. Hence, we should try to figure out what does $X$ measures.

• $q_t$ is the probablity of that a $t$ year old person is going to die before the person is able to become $t+1$ year old.

• $q_{t+y}$ is the probablity of that a $t+y$ year old person is going to die before the person is able to become $t+y+1$ year old.

• $1-q_{t+y}$ is the probablity of that an $t+y$ year old person is not going to die before the person is able to become $t+y+1$ year old.

• $1-q_{35+y}$ is the probablity of that an $35+y$ year old person is not going to die before the person is able to become $35+y+1$ year old.

• Hence $\prod_{y=0}^x(1-q_{35+y})$ is the probablity of that an $35$ year old person is not going to die before the person is able to become $35+x+1$ year old.

• Hence $1-\prod_{y=0}^x(1-q_{35+y})$ is the probablity of that an $35$ year old person is going to die before the person is able to become $35+x+1$ year old.

• That is, $P(X \le x)=1-\prod_{y=0}^x(1-q_{35+y})$ is the probablity of that an $35$ year old person is going to die before the person is able to become $35+x+1$ year old.
• $X$ measures how many more times a person of $35$ years old gets to celebrate his own birthday.

Though the original question didn't ask for the pmf, but if one is interested

\begin{align}P(X=x)&= F(x)-F(x-1) \\&= \left(1-\prod_{y=0}^x(1-q_{35+y}) \right) -\left(1-\prod_{y=0}^{x-1}(1-q_{35+y}) \right) \\&=\prod_{y=0}^{x-1}(1-q_{35+y}) - \prod_{y=0}^{x}(1-q_{35+y}) \\ &=q_{35+x} \prod_{y=0}^{x-1}(1-q_{35+y}) \end{align}

i.e. the person gets to celebrate $x-1$ birthday but not the next one.