Exponential distribution, am I doing this correctly? The time between successive cars on a certain road is exponentially distributed and the probability is $1/2$ that the next car will arrive within two minutes. Assume the time between and particular pair of cars is independent of the times between all other pairs of cars. 


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*What is the probability the next car will arrive within one minute?

*What is the expected time until the next car will arrive?
So I let T=the time (in minutes) between car arrivals
and this is exponentially distributed, so $f(t)=\lambda e^{-\lambda t}$, $t>0$. 
Also, I let N(t)= the number of car arrivals in a time interval t and this has a Poisson distribution.
So to find $\lambda$, I set N(2) (I plugged in $t=2$) equal to $\frac{1}{2}$ to solve for $\lambda$, then use that to find the probability for part $1$...is that correct?
 A: I feel as though you can solve this just by sticking to the exponential distribution. 
You could solve for $\lambda$ with the following equation:
$$\frac{1}{2} = \int_{0}^2 \lambda e^{-\lambda t}dt$$ and then once you have that, you can solve for the expected time with $$E = \lambda\int_0^{\infty} t e^{-\lambda t} dt$$ using the definition of expected value.
A: It is not clear what you mean by $N(2)$. The number of arrivals in $2$ minutes indeed has Poisson distribution with parameter $2\lambda$, where $\lambda$ is the parameter of the exponential. What $\lambda$ we would get from your calculation can only be clear if the calculation is shown. If $X$ is the Poisson with parameter $2\lambda$, and you put $\Pr(X\ge 1)=\frac{1}{2}$, or equivalently $\Pr(X=0)=\frac{1}{2}$, you will get the right $\lambda$.  
In the answer below, we calculate $\lambda$, so that you can verify whether your procedure gave the right number. 
The interarrival time $T$ has exponential distribution, with parameter say $\lambda$. Thus 
$$\Pr(T\le 2)=\int_0^2 \lambda e^{-\lambda t}\,dt=1-e^{-2\lambda}.$$
We are told that this probability is $\frac{1}{2}$, and therefore 
$$1-e^{-2\lambda}=\frac{1}{2}.$$
Solve. We get $\lambda=\frac{\ln 2}{2}$. 
Now that we know $\lambda$, the computations for questions 1 and 2 are routine. 
